Gareth Price, 16, was found hanging in Lancaster Farms Young Offender Institution on 19 January. He was then taken to Royal Lancaster Infirmary where he died the following day. He had been sent to the institution on 6 September after being convicted of rape.
He was waiting to be sentenced.
Karl Lewis, 18,was also found hanging last week at Stoke Heath Young Offender Institution. He had been convicted of robbery.
The suicides are the latest in a series of deaths in custody. Last year, Adam Rickwood, 14, was found hanging in Hassockfield Secure Training Centre, County Durham, and Gareth Myatt, 15, died after being restrained in Rainsbrook Secure Training Centre, Warwickshire.
Deborah Coles, co-director of Inquest, which works with bereaved families, said: "The tragic death of yet another child in State custody means the Government must set up a comprehensive public inquiry to learn the lessons about the treatment of children in the criminal justice system."
A Home Office spokesman said that the Prisons and Probation Ombudsman would be conducting an inquiry into the case.
