  We compute the relativistic corrections to the color-singlet contribution to
gluon fragmentation into a J/psi at relative order v^4, making use of the
nonrelativistic QCD (NRQCD) factorization approach. The corresponding full-QCD
process exhibits infrared divergences that manifest themselves as single and
double poles in epsilon in 4-2epsilon dimensions. We isolate the
infrared-divergent contributions and treat them analytically. In the matching
of full QCD to NRQCD, the pole contributions are absorbed into long-distance
NRQCD matrix elements. The renormalizations of the ultraviolet divergences of
the long-distance NRQCD matrix elements involve Born and one-loop single-pole
counterterm contributions and Born double-pole counterterm contributions. While
the order-v^4 contribution enhances the J/psi hadroproduction rate for the
color-singlet channel substantially, this contribution is not important
numerically in comparison with the color-octet contributions. We also find that
the ^3P_J color-octet channel in the gluon fragmentation function contributes
to J/psi hadroproduction significantly in comparison with the complete
contribution of next-to-leading order in alpha_s in that channel.
