  We present SWAS observations of water vapor in two cold star-less clouds, B68
and Core D in rho Ophiuchus. Sensitive non-detections of the 1(10)-1(01)
transition of o-H2O are reported for each source. Both molecular cores have
been previously examined by detailed observations that have characterized the
physical structure. Using these rather well defined physical properties and a
Monte-Carlo radiation transfer model we have removed one of the largest
uncertainties from the abundance calculation and set the lowest water abundance
limit to date in cold low-mass molecular cores. These limits are < 3 x 10^{-8}
(relative to H2) and < 8 x 10^{-9} in B68 and rho Oph D, respectively. Such low
abundances confirm the general lack of ortho-water vapor in cold (T < 20 K)
cores. Provided that the ortho/para ratio of water is not near zero, these
limits are well below theoretical predictions and appear to support the
suggestion that most of the water in dense low-mass cores is frozen onto the
surfaces of cold dust grains.
