The rates of the 18F(p,α)15O and 18F(p,γ)19Ne reactions at T = 0.1-0.4 GK are necessary to understand the emission of positron-annihilation gamma rays expected from novae. Observations of such gamma rays by space-based telescopes would provide direct tests of nova models. These rates, however, are uncertain because of the unknown level structure of 19Ne above the 18F+p threshold. Missing levels in 19Ne were searched for by stopping a 24-MeV 18F beam in a thick polypropylene target, and scattered protons were detected in a silicon-strip detector. The 18F(p,p)18F excitation function has been measured from Ec.m. = 0.3 - 1.3 MeV covering most energies of astrophysical interest, and a significant discrepancy was observed with a recent compilation of 19Ne resonance parameters.