The $^{17}O(p,\alpha)^{14}N$ and $^{17}O(p,\gamma)^{18}F$ reactions are of major importance to hydrogen-burning nucleosynthesis in a number of different stellar sites. In particular, $^{17}O$ and $^{18}F$ nucleosynthesis in classical novae is strongly dependent on the thermonuclear rates of these two reactions. The previously estimated rate for $^{17}O(p,\alpha)^{14}N$ carries very large uncertainties in the temperature range of classical novae (T=0.01-0.4 GK), whereas a recent measurement has reduced the uncertainty of the $^{17}O(p,\gamma)^{18}F$ rate. We report on the observation of a previously undiscovered resonance at Ec.m.=183.3 keV in the $^{17}O(p,\alpha)^{14}N$ reaction, with a measured resonance strength $\omega\gammap\alpha=(1.6\pm0.2)×10^{-3}$ eV. We studied in the same experiment the $^{17}O(p,\gamma)^{18}F$ reaction by an activation method, and the resonance strength was found to amount to $\omega\gammap\gamma=(2.2\pm0.4)×10^{-6}$ eV. The excitation energy of the corresponding level in 18F was determined to be 5789.8$\pm$0.3 keV in a Doppler shift attenuation method measurement, which yielded a value of $\tau$<2.6 fs for the level lifetime. The $^{17}O(p,\alpha)^{14}N$ and $^{17}O(p,\gamma)^{18}F$ reaction rates were calculated using the measured resonance properties and reconsidering some previous analyses of the contributions of other levels or processes. The $^{17}O(p,\alpha)^{14}N$ rate is now well established below T=1.5 GK, with uncertainties reduced by orders of magnitude in the temperature range T=0.1-0.4 GK. The uncertainty in the $^{17}O(p,\gamma)^{18}F$ rate is somewhat larger because of remaining obscurities in the knowledge of the direct capture process. These new resonance properties have important consequences for $^{17}$O nucleosynthesis and gamma-ray emission of classical novae.Read less <