Entrepreneurs Cameron and Tyler Winklevoss chose State Street Corp. to help them launch a new exchange-traded fund based on the virtual currency bitcoin, according to a regulatory filing Tuesday.

The twin brothers are hoping the Securities and Exchange Commission approves their new fund, the Winklevoss Bitcoin Trust, after a process that has already gone on for more than three years. In its latest filing, a seventh amendment to its original filing, the brothers said State Street, based in Boston, would serve as transfer agent,...