We observed the accretion-driven millisecond X-ray pulsars XTE J0929–314 and XTE J1751–305 in their quiescent states using Chandra . From XTE J0929–314 we detected 22 source photons ( in the energy range 0.3–8 keV ) in \sim 24.4 ksec , resulting in a background-corrected time-averaged count rate of 9 \pm 2 \times 10 ^ { -4 } counts s ^ { -1 } . The small number of photons detected did not allow for a detailed spectral analysis of the quiescent spectrum , 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 time-averaged ( averaged over the whole observation ) X-ray spectrum , we obtain a power-law index of 1.8 ^ { +0.6 } _ { -0.5 } and an unabsorbed X-ray flux of 6 ^ { +4 } _ { -2 } \times 10 ^ { -15 } ergs s ^ { -1 } cm ^ { -2 } ( for the energy range 0.5–10 keV ) , resulting in a 0.5–10 keV X-ray luminosity of 7 ^ { +5 } _ { -2 } \times 10 ^ { 31 } ( d /10 kpc ) ^ { 2 } ergs s ^ { -1 } , with d the distance toward the source 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 \sim 43 ksec observation , with 0.5–10 keV flux upper limits between 0.2 and 2.7 \times 10 ^ { -14 } ergs s ^ { -1 } cm ^ { -2 } depending on assumed spectral shape , resulting in 0.5–10 keV luminosity upper limits of 0.2 – 2 \times 10 ^ { 32 } ( d /8 kpc ) ^ { 2 } ergs s ^ { -1 } . We compare our results with those obtained for other neutron-star X-ray transients in their quiescent state , and in particular with the quiescent properties of SAX J1808.4–3658 . 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 \times 10 ^ { 9 } { d \over { 10 ~ { } kpc } } and 3 - 7 \times 10 ^ { 8 } { d \over { 8 ~ { } kpc } } Gauss ( depending on assumed spectral shape of the quiescent spectrum ) , respectively .