Detection of nuclear-decay γ rays provides a sensitive thermometer of nova nucleosynthesis. The most intense γ-ray flux is thought to be annihilation radiation from the β^{+} decay of ^{18}F, which is destroyed prior to decay by the ^{18}F(p,α)^{15}O reaction. Estimates of ^{18}F production had been uncertain, however, because key near-threshold levels in the compound nucleus, ^{19}Ne, had yet to be identified. We report the first measurement of the ^{19}F(^{3}He,tγ)^{19}Ne reaction, in which the placement of two long-sought 3/2^{+} levels is suggested via triton-γ-γ coincidences. The precise determination of their resonance energies reduces the upper limit of the rate by a factor of 1.5-17 at nova temperatures and reduces the average uncertainty on the nova detection probability by a factor of 2.1.