The rate of the 18F(p,γ)19Ne reaction affects the final abundance of the γ-ray observable radioisotope 18F, produced in novae. However, no successful measurement of this reaction exists and the rate used is calculated from incomplete information on the contributing resonances. Of the two resonances thought to play a significant role, one has a radiative width estimated from the assumed analogue state in the mirror nucleus, 19F. The second does not have an analogue state assignment at all, resulting in an arbitrary radiative width being assumed. Here, we report the first successful direct measurement of the 18F(p,γ)^19Ne reaction. The strength of the 665 keV resonance (Ex=7.076 MeV) is found to be over an order of magnitude weaker than currently assumed in nova models. Reaction rate calculations show that this resonance therefore plays no significant role in the destruction of ^{18}F at any astrophysical energy.