We use the NRQCD factorization formalism to calculate the relativistic corrections to the fragmentation function for a gluon fragmenting into a spin-triplet S-wave heavy quarkonium. We make use of the gauge-invariant formulation of the fragmentation function of Collins and Soper. The color-octet contribution receives a large, negative relativistic correction, while the color-singlet contribution receives a large, positive relativistic correction. The considerable decrease in the color-octet

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... ribution requires a corresponding increase in the phenomenological value of the leading color-octet matrix element in order to maintain a fit to the Fermilab Tevatron data.