The girl and her family were detained at a border patrol site in Texas when she fell ill, officials say.
The agency said she was transported to a local hospital where she was pronounced dead on Wednesday.
It said it would investigate the incident and that the Department of Homeland Security had been notified.
The Honduran consul in the border town of McAllen, Texas, José Leonardo Navas, identified the girl as Panama-born Anadith Tanay Reyes Alvarez.
Mr Navas told the Associated Press that she was travelling to the US with her Honduran parents and her two older siblings, and had been born with heart problems. Her father reportedly told the consul that she had been operated on in Panama three years ago.Her death appears to be the first in US Border Patrol custody for several years.
But it is the second death of a child from Latin America in US government custody in as many weeks.
Last week a 17-year-old Honduran boy who arrived unaccompanied in the US died at a government shelter facility in Florida.
Source: Girl, 8, dies in US Border Patrol custody, officials say