Classical novae are explained as thermonuclear explosions on the surface of white dwarf stars accreting hydrogen-rich material from less evolved companions in binary star systems. These events occur frequently within our galaxy and have been proposed as significant contributors to the galactic abundance of 13C, 15N, 17/18O and 18/19F. The short-lived isotope 18F (t1/2 = 110 min) is of particular importance since it may provide a signature of novae events through the detection of 511 keVγ-ray emission following the β+ decay of a 18F nucleus. During classical novae the 17O(p,γ)18F reaction governs the production of 18F and affects the synthesis of the rare isotopes mentioned above. Prior to the present study, the 17O(p,γ)18F reaction rate was poorly determined owing to a lack of low-energy experimental data. The present work reports on the first accurate measurements of the resonant and non-resonant contributions to the 17O(p,γ)18F reaction cross section in the energy region relevant for classical novae. Measurements were performed at the Laboratory for Underground Nuclear Astrophysics (LUNA) accelerator facility of the Laboratori Nazionali del Gran Sasso (LNGS) in Italy. Here the γ-ray background is suppressed by up to 3 orders of magnitude, thus providing a unique environment for low-energy measurements of reaction cross sections. Prompt γ rays associated with the formation and decay of states in 18F were analysed to determine the resonant and non-resonant contributions to the reaction cross section. The total non-resonant S-factor was determined at energies between Ecm ≈ 200 - 370 keV and the strength of a key resonance at Ecm = 183 keV was obtained with the best precision to date. The uncertainty in the reaction rate is now sufficiently low to place firmer constraints on nucleosynthesis predictions from accurate models of novae.