We present Submillimeter Wave Astronomy Satellite (SWAS) observations of the 1_{10}-1_{01} transition of ortho-water at 557 GHz toward 12 molecular cloud cores. The water emission was detected in NGC 7538, Rho Oph A, NGC 2024, CRL 2591, W3, W3(OH), Mon R2, and W33, and was not detected in TMC-1, L134N, and B335. We also present a small map of the water emission in S140. Observations of the H_2^{18}O line were obtained toward S140 and NGC 7538, but no emission was detected. The abundance of ortho-water relative to H_2 in the giant molecular cloud cores was found to vary between 6x10^{-10} and 1x10^{-8}. Five of the cloud cores in our sample have previous water detections; however, in all cases the emission is thought to arise from hot cores with small angular extents. The water abundance estimated for the hot core gas is at least 100 times larger than in the gas probed by SWAS. The most stringent upper limit on the ortho-water abundance in dark clouds is provided in TMC-1, where the 3-sigma upper limit on the ortho-water fractional abundance is 7x10^{-8}.