We present a new technique for detecting scattered starlight from transiting, close-orbiting extrasolar giant planets (CEGPs) that has the virtues of simplicity, robustness, linearity, and model-independence. Given a series of stellar spectra obtained over various phases of the planetary orbit, the goal is to measure the strength of the component scattered by the planet relative to the component coming directly from the star. We use two complementary strategies, both of which rely on the predictable Doppler shifts of both components and on combining the results from many spectral lines and many exposures. In the first strategy, we identify segments of the stellar spectrum that are free of direct absorption lines and add them after Doppler-shifting into the planetary frame. In the second strategy, we compare the distribution of equivalent-width ratios of the scattered and direct components. Both strategies are calibrated with a ``null test in which scrambled Doppler shifts are applied to the spectral segments. As an illustrative test case, we apply our technique to spectra of HD 209458 taken when the planet was near opposition (with orbital phases ranging from 11 to 34$arcdeg$, where 0$arcdeg$ is at opposition), finding that the planet-to-star flux ratio is $(1.4 pm 2.9)times10^{-4}$ in the wavelength range 554$-$681 nm. This corresponds to a geometric albedo of $0.8 pm 1.6$, assuming the phase function of a Lambert sphere. Although the result is not statistically significant, the achieved sensitivity and relatively small volume of data upon which it is based are very encouraging for future ground-based spectroscopic studies of scattered light from transiting CEGP systems.
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