  We calculate the rate of hydrogen burning for neutron stars (NSs) with
hydrogen atmospheres and an underlying reservoir of nuclei capable of proton
capture. This burning occurs in the exponentially suppressed diffusive tail of
H that extends to the hotter depths of the envelope where protons are rapidly
captured. This process, which we call diffusive nuclear burning (DNB), can
change the H abundance at the NS photosphere on timescales as short as
$10^{2-4}$ years. In the absence of diffusion, the hydrogen at the photosphere
(where $T\approx 10^6 {\rm K}$ and $\rho\sim 0.1 {\rm g cm^{-2}}$) would last
for far longer than a Hubble time. Our work impacts the understanding of the
evolution of surface abundances of isolated NSs, which is important to their
thermal spectrum and their effective temperature-core temperature relation. In
this paper, we calculate the rate of H burning when the overall consumption
rate is controlled by the nuclear timescales, rather than diffusion timescales.
The immediate application is for H burning on millisecond radio pulsars and in
quiescence for the accreting NS Cen X-4. We will apply this work to young radio
pulsars and magnetars once we have incorporated the effects of strong
$B>10^{12} {\rm G}$ magnetic fields.
