We have used the Submillimeter Wave Astronomy Satellite to map the ground-state 1_{10}-1_{01} transition of ortho-water at 557 GHz in the Orion molecular cloud. Water emission was detected in Orion over an angular extent of about 20 arcmin, or nearly 3 pc. The water emission is relatively weak, with line widths (3-6 km s^{-1}) and V_{LSR} velocities (9-11 km s^{-1}) consistent with an origin in the cold gas of the molecular ridge. We find that the ortho-water abundance relative to H_2 in the extended gas in Orion varies between 1 and 8x10^{-8}, with an average of 3x10^{-8}. The absence of detectable narrow-line ortho-H_2^{18}O emission is used to set a 3-sigma upper limit on the relative ortho-water abundance of 7x10^{-8}.