We have measured fluxes or flux limits for 31 of the 79 cluster candidates in the Palomar Distant Cluster Survey ( PDCS ) using archival ROSAT/PSPC pointed observations . Our X-ray survey reaches a flux limit of \simeq 3 \times 10 ^ { -14 } erg s ^ { -1 } cm ^ { -2 } ( 0.4 - 2.0 keV ) , which corresponds to luminosities of L _ { x } \simeq 5 \times 10 ^ { 43 } erg s ^ { -1 } ( { H _ { o } } = 50 km s ^ { -1 } Mpc ^ { -1 } , { q _ { o } } = \frac { 1 } { 2 } ) , if we assume the PDCS estimated redshifts . Of the 31 cluster candidates , we detect six at a signal-to-noise greater than three . We estimate that 2.9 ^ { +3.3 } _ { -1.4 } ( 90 % confidence limits ) of these six detections are a result of X-ray emission from objects unrelated to the PDCS cluster candidates . The net surface density of X-ray emitting cluster candidates in our survey , 1.71 ^ { +0.91 } _ { -2.19 } clusters deg ^ { -2 } , agrees with that of other , X-ray selected , surveys . It is possible , given the large error on our contamination rate , that we have not detected X-ray emission from any of our observed PDCS cluster candidates . We find no statistically significant difference between the X-ray luminosities of PDCS cluster candidates and those of Abell clusters of similar optical richness . This suggests that the PDCS contains objects at high redshift similar to the low redshift clusters in the Abell catalogs . We show that the PDCS cluster candidates are not bright X-ray sources , the average luminosity of the six detected candidates is only \bar { L _ { x } } = 0.9 \times 10 ^ { 44 } erg s ^ { -1 } ( 0.4-2.0 keV ) . This finding is in agreement with previous X-ray studies of high redshift , optically selected , rich clusters of galaxies .