  (Abridge) We observed the accreting millisecond X-ray pulsars XTE J0929-314
and XTE J1751-305 in their quiescent states using Chandra. From XTE J0929-314
we detected 22 photons (0.3-8 keV) in 24.4 ksec, resulting in a count rate of 9
x 10^{-4} c/s. The small number of photons detected did not allow for a
detailed spectral analysis, but we can demonstrate that the spectrum is harder
than simple thermal emission which is what is usually presumed to arise from a
cooling neutron star that has been heated during the outbursts. Assuming a
power-law model for the spectrum, we obtain a power-law index of ~1.8 and an
unabsorbed flux of 6 x 10^{-15} ergs/s/cm^2 (0.5-10 keV), resulting in a
luminosity of 7 x 10^{31} (d/10 kpc)^2 ergs/s, with d in kpc. No thermal
component could be detected; such a component contributed at most 30% to the
0.5-10 keV flux. Variability in the count rate of XTE J0929-314 was observed at
the 95% confidence level. We did not conclusively detect XTE J1751-305 in our
43 ksec observation, with 0.5-10 keV flux upper limits between 0.2 and 2.7 x
10^{-14} ergs/s/cm^2 depending on assumed spectral shape, resulting in
luminosity upper limits of 0.2 - 2 x 10^{32} (d/8 kpc)^2 ergs/s. We compare our
results with those obtained for other neutron-star X-ray transients in their
quiescent state. Using simple accretion disk physics in combination with our
measured quiescent luminosity of XTE J0929-314 and the luminosity upper limits
of XTE J1751-305, and the known spin frequency of the neutron stars, we could
constrain the magnetic field of the neutron stars in XTE J0929-314 and XTE
J1751-305 to be less than 3 x 10^9 (d/10 kpc) and 3 - 7 x 10^8 (d/8 kpc) Gauss
(depending on assumed spectral shape of the quiescent spectrum), respectively.
