We report resolved photometry of the primary and secondary components of 23 transneptunian binaries obtained with the Hubble Space Telescope. V-I colors of the components range from 0.7 to 1.5 with a median uncertainty of 0.06 magnitudes. The colors of the primaries and secondaries are correlated with a Spearman rank correlation probability of 99.99991%, 5 sigma for a normal distribution. Fits to the primary vs. secondary colors are identical to within measurement uncertainties. The color range of binaries as a group is indistinguishable from that of the larger population of apparently single transneptunian objects. Whatever mechanism produced the colors of apparently single TNOs acted equally on binary systems. The most likely explanation is that the colors of transneptunian objects and binaries alike are primordial and indicative of their origin in a locally homogeneous, globally heterogeneous protoplanetary disk.