Priscilla Nicole Maldonado was taken to University Medical Center at about 8 p.m., where a hospital spokesman said she appears to be fine but was kept overnight for observation.
Police said the infant was found alone in a car seat beneath a carport in 104-degree weather. It was not clear how long she had been outside before authorities arrived.
"She's doing good," Erica Ysasaga, the baby's mother, told CBS News' The Early Show. Priscilla was sleeping in her arms as she spoke.
Ysasaga told The Early Show the past 24-hours were "exhausting, stressful, just wondering about my baby...and finally (there was) excitement."
Early Tuesday, police told CBSNews.com that the suspect "has made a full statement" to authorities, has no relationship to the victim, and the incident remains under investigation.
Lt. Roy Bassett said police rescued the child after a caller from Amarillo suggested they contact a man in Lubbock. Police called the man and asked if his wife had recently had a baby. Bassett says the man replied that she had, but his answers had "obvious flaws."
Police say Stephanie Lynn Anderson Jones, 33, of Amarillo, Texas, is being held on charges of kidnapping and endangering the welfare of a child.
CBS News Affiliate KLBK-TV quotes Bassett as saying that police are also looking at other "persons of interest."
Detectives have been questioning Stephanie Jones as well as her husband.
Bassett says she closely matches the description the baby's family had provided of the woman who had visited them several times in the hospital last week and disappeared with their daughter Sunday after mother and child had gone home.
Police Chief Claude Jones said the woman was taken into custody as she drove into the apartment complex. He said she led officers to the baby.
The abduction happened seven months after the death of Ysasaga's and Jesse Maldonado's 2-month-old daughter, who reportedly choked to death while being burped in October.
Bassett said late Monday that police ruled out foul play in the older baby's death. Early Tuesday, police had no comment on the details of that case.