In this Paper we report on our analysis of three Chandra observations of the accretion–powered millisecond X–ray pulsar IGR J00291+5934 obtained during the late stages of the 2004 outburst . We also report the serendipitous detection of the source in quiescence by ROSAT during MJD 48830–48839 ( July 26–August 4 , 1992 ) . The detected 0.3–10 keV source count rates varied significantly between the Chandra observations from ( 7.2 \pm 1.2 ) \times 10 ^ { -3 } , ( 6.8 \pm 0.9 ) \times 10 ^ { -3 } , and ( 1.4 \pm 0.1 ) \times 10 ^ { -2 } counts per second for the first , second , and third Chandra observation , on MJD 53371.88 ( Jan. 1 , 2005 ) , 53383.99 ( Jan. 13 , 2005 ) , and 53407.57 ( Feb. 6 , 2005 ) , respectively . The count rate for the third observation is 2.0 \pm 0.4 times as high as that of the average of the first two observations . The unabsorbed 0.5–10 keV source flux for the best–fit power–law model to the source spectrum was ( 7.9 \pm 2.5 ) \times 10 ^ { -14 } erg cm ^ { -2 } s ^ { -1 } , ( 7.3 \pm 2.0 ) \times 10 ^ { -14 } erg cm ^ { -2 } s ^ { -1 } , and ( 1.17 \pm 0.22 ) \times 10 ^ { -13 } erg cm ^ { -2 } s ^ { -1 } for the first , second , and third Chandra observation , respectively . We find that this source flux is consistent with that found by ROSAT [ \approx ( 5.4 \pm 2.4 ) \times 10 ^ { -14 } erg cm ^ { -2 } s ^ { -1 } ] . Under the assumption that the interstellar extinction , N _ { H } , does not vary between the observations , we find that the blackbody temperature during the second Chandra observation is significantly higher than that during the first and third observation . Furthermore , the effective temperature of the neutron star derived from fitting an absorbed blackbody or neutron star atmosphere model to the data is rather high in comparison with many other neutron star soft X–ray transients in quiescence , even during the first and third observation . If we assume that the source quiescent luminosity is similar to that measured for two other accretion powered millisecond pulsars in quiescence , the distance to IGR J00291+5934 is 2.6–3.6 kpc .