Irving police spokesman David Tull identified the dead child as Zain Akhter. In a statement issued late Monday, he said that Zain's sister, Faryaal Akhter, was in "very critical" condition Monday night at Children's Medical Center at Dallas.
Tull said the 30-year-old mother remains in custody without bond.
In the statement, Tull says the department received a call from the mother about 5 p.m. Monday, in which she said she had harmed her children.
Officers arrived to find both children apparently strangled and unconscious.