We have observed X-ray shadowing by nine nearby high-latitude molecula
r clouds with the ROSAT PSPC. The 1/4 keV emission from the Local Bubb
le is used to calculate an average electron density, n(e), toward all
of our sample clouds. Given the uniformity of emission within the Loca
l Bubble, and previously determined cloud distances, one can determine
whether the clouds are within or without the Local Bubble. We confirm
that three of the sample clouds, MBM 12, MBM 16, and possibly MBM 20,
which have been thought to be within the Local Bubble, are actually w
ithin. A combination of 1/4 keV and n(e) measures suggest that MBM 6 a
nd LDN 1563 may also lie within the Local Bubble or its boundary layer
. The shadows cast by the clouds at 3/4 keV and 1.5 keV imply little i
f any foreground emission in those bands, and produce lower limits for
the distance to the hotter emission component responsible for the obs
erved flux. The results are in agreement with previous estimates for t
he distances to the Loop I Bubble and the Eridion Bubble.