  We examine a number of evolutionary scenarios for the recently discovered
class of accretion-powered millisecond X-ray pulsars in ultracompact binaries,
including XTE J0929-314 and XTE J1751-305, with orbital periods of 43.6 and
42.4 minutes, respectively. We focus on a particular scenario that can
naturally explain the present-day properties of these systems. This model
invokes a donor star that was either very close to the TAMS (i.e.,
main-sequence turnoff) at the onset of mass transfer or had sufficient time to
evolve during the mass-transfer phase. We have run a systematic set of detailed
binary evolution calculations with a wide range of initial donor masses and
degrees of (nuclear) evolution at the onset of mass transfer. In general, the
models whose evolutionary tracks result in the best fits to these ultracompact
binaries start mass transfer with orbital periods of ~15 hr, then decrease to a
minimum orbital period of less than or about 40 minutes, and finally evolve
back up to about 43 minutes. We also carry out a probability analysis based on
the measured mass functions of XTE J0929-314 and XTE J1751-305, and combine
this with the results of our binary evolution models and find that the donor
stars currently have masses in the range of about 0.012 - 0.025 solar masses,
and radii of about 0.042 - 0.055 solar radii, and that these radii are likely
to be factors of about 1.1 - 1.3 times larger than the corresponding
zero-temperature ones. We also find that the interiors of the donors are
largely composed of helium and that the surface hydrogen abundances are almost
certainly less than 10% (by mass).
