The ReChord Trial is studying the safety and effectiveness of the NeoChord DS1000 system to treat patients with degenerative mitral valve regurgitation. The study is designed as a randomized control trial. Up to 450 participants will be randomly put into one of two different groups in up to 20 enrollment centers. The control group will receive a standard open surgical repair in a stopped heart. The device group will receive a beating heart mitral valve repair with the NeoChord DS1000 system.