Hydrogen burning of the oxygen isotopes takes place in low-mass stars, asymptotic giant branch stars, and classical novae. Observations of oxygen elemental and isotopic abundances in stellar spectra or in presolar grains provide strong constraints for stellar models if reliable thermonuclear reaction rates for hydrogen burning of oxygen are available. We present the results of a new measurement of the 17O(p,γ)18F reaction in the laboratory bombarding energy range of 170–530 keV. The measurement is performed with significantly higher beam intensities (Imax ≈ 2 mA) compared to previous work and by employing a sophisticated γ-ray coincidence spectrometer. We measured the cross section at much lower energies than previous in-beam experiments. We also apply a novel data-analysis technique that is based on the decomposition of different contributions to the measured pulse-height spectrum. Our measured strengths of the low-energy resonances amount to ωγ_pres(193keV)=(1.86±0.13)×10−6 eV and ωγ_pres(518keV)=(13.70±0.96)×10−3 eV. For the direct capture S factor at zero energy, we find a value of S_pres,DC(0) = 4.82±0.41 keV b. We also present new thermonuclear rates for the 17O+p reactions, taking into account all consistent results from previous measurements.