We completed a measurement of the Stark shift within the In-115 6s(1/2). -> 6p(1/2) excited-state transition using two-step laser spectroscopy in an indium atomic beam. By combining this measurement with recent experimental results we determine the scalar polarizability alpha(0) of the 6p(1/2) state to be 7683 +/- 43a(0)(3) (in atomic units), a result which agrees very well with recent theoretical calculations. In this experiment, one laser, stabilized to the 5p(1/2). -> 6s(1/2) 410 nm transition, was directed transversely to the atomic beam, while a second, overlapping laser was scanned across the 1343 nm 6s(1/2) -> 6p(1/2) transition. We utilized two-tone frequency-modulation spectroscopy of the infrared laser beam to measure the second-step absorption in the interaction region, where the optical depth is less than 10(-3). In the course of our experimental work we also determined the hyperfine splitting within the 6p(1/2) state, improving upon the precision of an existing measurement.