We discuss the discovery and characterization of the circumbinary planet Kepler-38b. The stellar binary is single-lined, with a period of 18.8 days, and consists of a moderately evolved main-sequence star (M-A = 0.949+/-0.059 M-circle dot and R-A = 1.757+/-0.034 R-circle dot) paired with a low-mass star (M-B = 0.249+/-0.010 M-circle dot and R-B = 0.2724+/-0.0053 R-circle dot) in a mildly eccentric (e = 0.103) orbit. A total of eight transits due to a circumbinary planet crossing the primary star were identified in the Kepler light curve (using Kepler Quarters 1-11), from which a planetary period of 105.595+/-0.053 days can be established. A photometric dynamical model fit to the radial velocity curve and Kepler light curve yields a planetary radius of 4.35+/-0.11 R-circle plus, or equivalently 1.12+/-0.03 R-Nep. Since the planet is not sufficiently massive to observably alter the orbit of the binary from Keplerian motion, we can only place an upper limit on the mass of the planet of 122 M-circle dot(7.11 M-Nep or equivalently 0.384 M-Jup) at 95% confidence. This upper limit should decrease as more Kepler data become available.