The patient, currently in serious condition, is at Texas Health Presbyterian Hospital.
Officials at the hospital admitted on Wednesday the patient initially went to a Dallas emergency room last week but was sent home, despite telling a nurse that he had been in disease-ravaged West Africa.
He returned to the hospital two days later in an ambulance.
CDC Director Tom Frieden said that the patient did not show any symptoms when leaving Liberia on Sept. 19 or entering the U.S. on Sept. 20.
United Airlines said Wednesday it believes the patient spent part of his flights out of Africa to the U.S. on United flights. United believes he was on a Brussels to Washington Dulles Flight 951 and Dulles to Dallas-Fort Worth Flight 822 on Sept. 20. Officials said there is no risk to anyone on those flights.
The patient, identified by relatives as Thomas Eric Duncan, was staying at the Ivy Apartments on Fair Oaks Avenue in Dallas. The complex is just southeast of the hospital where he's being treated.
Duncan first sought treatment on Sept. 26. He complained of an illness and told nurses he was visiting from Liberia. But hospital officials said that information was not passed along to other personnel and he was released with antibiotics.
The decision by Texas Health Presbyterian Hospital to release the patient, who had recently arrived from Liberia, could have put others at risk of exposure to Ebola before the man went back to the ER a couple of days later, when his condition worsened.
"Regretfully, that information was not communicated throughout the full team, and as a result, the full import of that information wasn't factored into clinical decision making," said Texas Health Resources' Dr. Mark Lester.
Instead, Duncan was diagnosed with a low-risk infection and sent home. He was prescribed antibiotics, according to his sister, Mai Wureh.
He returned to the hospital on Sept. 28 as his condition worsened and was initially in critical condition, but his condition was upgraded Wednesday to serious condition.
Dr. Edward Goodman with Texas Health Presbyterian Hospital wouldn't comment on drugs being used to treat the patient, but he said that there isn't any more ZMapp available, which is an experimental drug that was used on two previous Ebola patients.
However, Frieden said the hospital is discussing if experimental treatments would be appropriate.
The City of Dallas says Duncan had just moved to North Texas.
The New York Times reports Duncan quit his shipping company job in Monrovia because he'd gotten a visa to the U.S. and was coming to be with his family.
The Times says that while in Monrovia on Sept. 15, his landlord's daughter got very sick and Duncan helped carry the woman to the hospital.
She died the next day.
Then, the landlord's son became ill and he died.
At least two others in the same neighborhood who may have had contact with the woman also died.
At an address listed for the family in Kannapolis, North Carolina, a man who identified himself as Duncan's brother told reporters Wednesday the family was "just stressed out."
RISK TO OTHERS AND PRECAUTIONS
Ebola is spread through direct contact (through broken skin or mucous membranes) with blood or bodily fluids of a person who is sick with Ebola, objects (like needles and syringes) that have been contaminated with the virus and infected animals.
Dallas Fire-Rescue Assistant Chief Norman Seals said Tuesday night that the Ebola patient was transported to the hospital by three crew members from Fire Station 37, who are now in reverse isolation at home. That means they've been asked to stay home from work for 21 days, will be checked on daily and will have their temperatures monitored.
The ambulance that the Ebola patient traveled in is being quarantined. It was used for 48 hours after the patient was transported to the hospital.
Dallas Fire-Rescue is waiting on experts to guide it on how to properly sanitize and decontaminate the ambulance.
There was no risk to anyone on the airplane because the man had no symptoms at the time of the flight, Frieden said.
Several Dallas Independent School District children possibly had contact with the patient.
Superintendent Mike Miles said the students attend Tasby Middle School, Conrad High School, Hotchkiss Elementary School, Jack Lowe Elementary School and Dan D. Rogers Elementary School. They were in school earlier this week, but are now at home and being monitored.
Additionally, Dallas ISD officials have said that health experts and extra custodians will be at schools on Thursday where the children who may have come into contact with Duncan attend.
CDC officials and doctors at the press conference emphasized that they believe the patient's diagnosis will be controlled, and that the risk for an outbreak in Dallas is low.
"The bottom line here is that I have no doubt we will control this importation, or this case of Ebola, so that it does not spread widely in this country," Frieden said.
Both the hospital and health officials said strict federal isolation and testing guidelines are being followed.
"Unless you have exchanged bodily fluids with this individual this is a low risk for people in Dallas County. So we want to ease the fear. We're talking about bodily fluids transmission, not just sitting right here together that being an issue," said Dallas County Health Director Zach Thompson.
The CDC has sent a team to Dallas from its Atlanta headquarters Part of the team landed Tuesday night in North Texas.
That team, of mostly epidemiologists, will help local and state health departments with tracing the disease to anyone who may have had contact with the patient, and the team will handle other parts of the investigation.
However, officials said Tuesday there are no other suspected cases in Texas.
"It is certainly possible that someone who had contact with this individual, a family member or other individual, could develop Ebola in the coming weeks," said Frieden. "But there is no doubt in my mind that we will stop it here.
The City of Dallas has also opened its Emergency Operations Center in response to the Ebola diagnosis. It's at level 2; high readiness status and will help coordinate operations between the city, county, state and the CDC.
Dallas Mayor Mike Rawlings says he has full faith in the staff at Presbyterian Hospital to handle the Ebola diagnosis.
He says the state of Texas was tested in August as to how it would treat Ebola patients, and the plan enacted by Dallas County then is now in place.
"We have mobilized our resources to make sure everybody is safe," said Rawlings. "I understand people's fears, but I think it's very, very hard to transmit if the proper procedures are in place."
Four American aid workers who became infected in West Africa have been flown back to the U.S. for treatment after they became sick. They were cared for in special isolation facilities at hospitals in Atlanta and Nebraska. Three have recovered.
Also, a U.S. doctor exposed to the virus in Sierra Leone is under observation in a similar facility at the National Institutes of Health.
The U.S. has only four such isolation units. Asked whether the Texas patient would be moved to one of those specialty facilities, Frieden said there was no need and virtually any hospital can provide the proper care and infection control.
Thompson said that unlike Africa, Dallas County has the infrastructure needed to deal with Ebola.
Since the summer months, U.S. health officials have been preparing for the possibility that an individual traveler could unknowingly arrive with the infection. Health authorities have advised hospitals on how to prevent the virus from spreading within their facilities.
People boarding planes in the outbreak zone are checked for fever, but that does not guarantee that an infected person won't get through.
The CDC has said that 12 other people in the U.S. have been tested for Ebola since July 27. Those tests came back negative.
The World Health Organization says the Ebola outbreak in West Africa has killed more than 3,000 people.
More on Ebola:
The Associated Press contributed to this report.