We have obtained high resolution images of the central regions of 14 reverberation-mapped active galactic nuclei (AGN) using the Hubble Space Telescope Advanced Camera for Surveys High Resolution Camera to account for host-galaxy starlight contamination of measured AGN luminosities. We measure the host-galaxy starlight contribution to the continuum luminosity at 5100 A through the typical ground-based slit position and geometry used in the reverberation-mapping campaigns. We find that removing the starlight contribution results in a significant correction to the luminosity of each AGN, both for lower luminosity sources, as expected, but also for the higher luminosity sources such as the PG quasars. After accounting for the host galaxy starlight, we revisit the well-known broad-line region radius--luminosity relationship for nearby AGN. We find the power-law slope of the relationship for the H beta line to be 0.518 +/- 0.039, shallower than previously reported and consistent with the slope of 0.5 expected from the naive theoretical assumption that all AGN have, on average, the same ionizing spectrum and the same ionization parameter and gas density in the H beta line-emitting region. Comment: 27 pages, 5 tables, 4 figures, accepted to ApJ; full resolution figures available at http://www.astronomy.ohio-state.edu/~bentz/astroph0602412.html