With veteran sidekick Gerald Sensabaugh (calf) and Matt Johnson (hamstring) also out against the Bucs, Mana Silva was thrust into significant playing time.
It's a big setback for Church, an undrafted rookie in 2010 who finally earned the starting job this season.
"It's disappointing but I have to stay upbeat and positive for the team," Church told the Fort Worth Star-Telegram. "I will continue to do that."
Church said he might have been vulnerable to the Achilles injury while playing through a quad bruise suffered to the same leg at Seattle in Week 2.
"It was run play," Church said. "I think it was a freak accident. I went to help on the run and soon as I planted it popped."
-- The Sports Xchange