  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.
