  We present the results of a HST WFPC2 search for substellar companions to
members of the star-forming cluster IC 348. The detection limits for companions
to low-mass stars and brown dwarfs in the PC images are dm_791=0, 2.5, and 5.5
at separations of 0.05", 0.1", and 0.3", respectively, which correspond to
M_2/M_1=1, 0.3, and 0.1 at 15, 30, and 90 AU. Meanwhile, for heavily saturated
solar-mass primaries in the WFC images, the limits are dm_791=0 and 6
(M_2/M_1=1 and 0.04) at 0.2" and 0.4". The sky limiting magnitude of m_791~26
at large separations from a primary corresponds to a mass of ~0.006 M_sun. For
the two faintest candidate companions appearing in these data, we have obtained
0.8-2.5 um spectra with SpeX at the IRTF, which we classify as cluster members
with spectral types near M6, corresponding to masses of ~0.1 M_sun. Thus, no
probable substellar companions are detected in this survey. The frequencies of
stellar and substellar companions within 0.4-5" (120-1600 AU) from low-mass
stars in IC 348 agree with measurements in the field. The factor of ~3-10
deficiency in brown dwarfs relative to stars among companions at wide
separations in IC 348 and across the much larger range of separations probed
for field stars is equal within the uncertainties to the deficiency in brown
dwarfs in measurements of mass functions of isolated objects. In other words,
when defined relative to stars, the brown dwarf "desert" among companions is
also present among isolated objects, which is expected if stellar and
substellar companions form in the same manner as their free-floating
counterparts. Meanwhile, among the 14 substellar primaries in our survey of IC
348, no companions are detected. This absence of wide binary brown dwarfs is
statistically consistent with the frequency of wide binary stars in IC 348.
