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Neutron stars, with their strong surface gravity, have interestingly short timescales for the sedimentation of heavy elements. Motivated by observations of Type I X-ray bursts from sources with extremely low persistent accretion luminosities, L_X<10³⁶ ergs/s (≃0.01L_Edd), we study how sedimentation affects the distribution of isotopes and the ignition of H and He in the envelope of an accreting neutron star. For local mass accretion rates m{dot}≲10^–2m{dot}_Edd (for which the ignition of H is unstable), where m{dot}_Edd=8.8x10⁴ g/cm2/s, the helium and CNO elements sediment out of the accreted fuel before reaching a temperature at which H would ignite. Using one-zone calculations of the thermonuclear burning, we find a range of accretion rates for which the unstable H ignition does not trigger unstable He burning. This range depends on the emergent flux from reactions in the deep neutron star crust; for F=0.1 MeV(m{dot}/m_u), the range is 3x10^–3m{dot}_Edd≲m{dot}≲10^–2m{dot}_Edd. We speculate that sources accreting in this range would build up a massive He layer that would later produce an energetic and long X-ray burst. At mass accretion rates lower than this range, we find that the H flash leads to a strong mixed H/He flash. Surprisingly, even at accretion rates m{dot}≳0.1m{dot}_Edd, although the H and He do not completely segregate, the H abundance at the base of the accumulated layer is still reduced. While following the evolution of the X-ray burst is beyond the scope of this introductory paper, we note that the reduced proton-to-seed ratio favors the production of ¹²C–an important ingredient for subsequent superbursts.