Noam Shalit, his father, would not quote directly from the letter, but said his son also wrote of poor health and dreams of returning home.
Excerpts from another letter from Shalit was released in September 2006. An audio tape was released last June.
'Gesture to Carter'
Osama al-Muzaini, a senior Hamas official, said the letter was released in a gesture to Carter who met with the group's leaders during a trip to the region in April.
"The letter was sent in a goodwill gesture to former President Jimmy Carter and to indicate Hamas's seriousness in its desire to reach a prisoner swap deal that will close the Shalit file and fulfil the demands of the Palestinian groups who abducted Shalit."
Hamas' military wing and two other Palestinian groups claimed responsibility for kidnapping Shalit.
They have demanded the release of about 1,400 Palestinian prisoners by Israel in exchange for releasing Shalit.
Hamas took control of power in Gaza from its Palestinian rival Fatah in June 2007.
Egypt is currently attempting to mediate a peace agreement between Palestinian groups and Israel, which is being held up due to non-agreement on conditions.