  In decays of the $Z^0$, the dominant mechanism for the direct production of
charmonium states is the decay of the $Z^0$ into a charm quark or antiquark
followed by its fragmentation into the charmonium state. We calculate the
fragmentation functions describing the splitting of charm quarks into S-wave
charmonium states to leading order in the QCD coupling constant. Leading
logarithms of $M_Z/m_c$ are summed up using Altarelli-Parisi evolution
equations. Our analytic result agrees with the complete leading order
calculation of the rate for $Z^0 \rightarrow \psi c {\bar c}$. We also use our
fragmentation functions to calculate the production rate of heavy quarkonium
states in $W^\pm$, top quark, and Higgs decays.
