  We derive a factorization theorem for the Higgs-boson production amplitude in
gluon-gluon fusion induced by a light-quark loop, working at next-to-leading
power in soft-collinear effective theory. The factorization is structurally
similar to that obtained for the $h\to\gamma\gamma$ decay amplitude induced by
a light-quark loop, but additional complications arise because of external
color charges. We show how the refactorization-based subtraction scheme
developed in previous work leads to a factorization theorem free of endpoint
divergences. We use renormalization-group techniques to predict the
logarithmically enhanced terms in the three-loop $gg\to h$ form factor of order
$\alpha_s^3\ln^k(-M_h^2/m_b^2)$ with $k=6,5,4,3$. We also resum the first three
towers of leading logarithms, $\alpha_s^n\ln^{2n-k}(-M_h^2/m_b^2)$ with
$k=0,1,2$, to all orders of perturbation theory.
