The father and son were not identified, and the agency statement said it will release more details "as available and appropriate".
A mother migrating from Honduras holds her one-year-old child as surrendering to Border Patrol agents after illegally crossing the Texas border.
This is the second immigrant child this month to die in custody.
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In the second case, an agent noticed Monday that the child had become ill. Agency guidelines say immigrants generally shouldn't be detained for more than 72 hours in CBP holding facilities, which are usually smaller and have fewer services than ICE's detention centers.
The border agency has not yet said when the father and son entered the United States or how long they were detained, saying only in its statement that the boy had been "previously apprehended". Guatemalan officials are meeting with the boy's father and will speak with any family members who are in Guatemala, according to the news release.
Customs and Border Protection said the child's official cause of death has not been determined.
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CBP announced new notification procedures in response to Jakelin's death, which was not revealed until several days later.
In an official statement, the Department of Homeland Security reported that a US border patrol agent noticed the minor exhibiting symptoms of illness on Monday and transported the boy, with his father, to a local hospital. In a news conference earlier this month, a migrant advocate said the girl's father, 29-year-old Nery Caal, had told him that Jakelin was healthy and had no pre-existing conditions.
The family's attorneys have said that it was "unacceptable" to have Jakelin's father sign a document in a language that he didn't understand.
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The girl's remains were then returned to her family's remote village in Guatemala.