The $^{17}$O(p,$alpha$)$^{14}$N reaction plays a key role in various astrophysical scenarios, from asymptotic giant branch stars to classical novae. It affects the synthesis of rare isotopes such as $^{17}$O and $^{18}$F, which can provide constraints on astrophysical models. A new direct determination of the $E_{rm R}~=~64.5$~keV resonance strength performed at the Laboratory for Underground Nuclear Astrophysics accelerator has led to the most accurate value to date, $omegagamma = 10.0 pm 1.4_{rm stat} pm 0.7_{rm syst}$~neV, thanks to a significant background reduction underground and generally improved experimental conditions. The (bare) proton partial width of the corresponding state at $E_{rm x} = 5672$~keV in $^{18}$F is $Gamma_{rm p} = 35 pm 5_{rm stat} pm 3_{rm syst}$~neV. This width is about a factor of 2 higher than previously estimated thus leading to a factor of 2 increase in the $^{17}$O(p,$alpha$)$^{14}$N reaction rate at astrophysical temperatures relevant to shell hydrogen-burning in red giant and asymptotic giant branch stars. The new rate implies lower $^{17}$O/$^{16}$O ratios, with important implications on the interpretation of astrophysical observables from these stars.