  We present a fully analytic calculation of the Higgs boson transverse
momentum and rapidity distributions, for nonzero Higgs $p_\perp$, at
next-to-leading order in the infinite-top-mass approximation. We separate the
cross section into a part that contains the dominant soft, virtual, collinear,
and small-$p_\perp$-enhanced contributions, and the remainder, which is
organized by the contributions due to different parton helicities. We use this
cross section to investigate analytically the small-$p_\perp$ limit and compare
with the expectation from the resummation of large logarithms of the type
$\ln{m_H/p_\perp}$. We also compute numerically the cross section at moderate
$p_\perp$ where a fixed-order calculation is reliable. We find a $K$-factor
that varies from $\approx1.6-1.8$, and a reduction in the scale dependence, as
compared to leading order. Our analysis suggests that the contribution of
current parton distributions to the total uncertainty on this cross section at
the LHC is probably less than that due to uncalculated higher orders.
