We have used the CLEO-II detector at the Cornell Electron Storage Ringe (CESR) to study the inclusive production of charmonium mesons in a sample of 2.15 million BB events. We find inclusive branching fractions of (1.120.040.06)% for BJ/X, (0.340.040.03)% for BX, and (0.400.060.04)% for Bc1X. We also find some evidence for the inclusive production of c2, and set an upper limit for the branching fraction of the inclusive decay BcX of 0.9% at 90% confidence level. Momentum spectra for inclusive J/, and c1 production are presented. These measurements are compared to theoretical calculations.