  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$^{-1}$, $q_o$ = ${1/2}$), 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 $\bar{L_x}=0.9\times10^{44}$ 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.
