We present results of a high angular resolution survey of massive OB stars in the Cygnus OB2 association that we conducted with the fine guidance sensor 1R (FGS1r) on the Hubble Space Telescope. FGS1r is able to resolve binary systems with a magnitude difference ΔV < 4 down to separations as small as 0.''01. The sample includes 58 of the brighter members of Cyg OB2, one of the closest examples of an environment containing a large number of very young and massive stars. We resolved binary companions for 12 targets and confirmed the triple nature of one other target, and we offer evidence of marginally resolved companions for two additional stars. We confirm the binary nature of 11 of these systems from complementary adaptive optics imaging observations. The overall binary frequency in our study is 22% to 26% corresponding to orbital periods ranging from 20 to 20,000 yr. When combined with the known short-period spectroscopic binaries, the results support the hypothesis that the binary fraction among massive stars is >60%. One of the new discoveries is a companion to the hypergiant star MT 304 = Cyg OB2-12, and future measurements of orbital motion should provide mass estimates for this very luminous star.