Exoplanetary transmission spectroscopy in the near-infrared using the Hubble Space Telescope(HST) NICMOS is currently ambiguous because different observational groups claim different results from the same data, depending on their analysis methodologies. Spatial scanning with HST/WFC3 provides an opportunity to resolve this ambiguity. We here report WFC3 spectroscopy of the giant planets HD 209458b and XO-1b in transit, using spatial scanning mode for maximum photon-collecting efficiency. We introduce an analysis technique that derives the exoplanetary transmission spectrum without the necessity of explicitly decorrelating instrumental effects, and achieves nearly photon-limited precision even at the high flux levels collected in spatial scan mode. Our errors are within 6% (XO-1) and 26% (HD 209458b) of the photon-limit at a resolving power of λ/δλ ~ 70, and are better than 0.01% per spectral channel. Both planets exhibit water absorption of approximately 200 ppm at the water peak near 1.38 μm. Our result for XO-1b contradicts the much larger absorption derived from NICMOS spectroscopy. The weak water absorption we measure for HD 209458b is reminiscent of the weakness of sodium absorption in the first transmission spectroscopy of an exoplanet atmosphere by Charbonneau et al. Model atmospheres having uniformly distributed extra opacity of 0.012 cm2 g–1 account approximately for both our water measurement and the sodium absorption. Our results for HD 209458b support the picture advocated by Pont et al. in which weak molecular absorptions are superposed on a transmission spectrum that is dominated by continuous opacity due to haze and/or dust. However, the extra opacity needed for HD 209458b is grayer than for HD 189733b, with a weaker Rayleigh component.