Classical novae are expected to contribute to the 1809-keV Galactic $gamma$-ray emission by producing its precursor $^{26}$Al, but the yield depends on the thermonuclear rate of the unmeasured $^{25}$Al($p,gamma$)$^{26}$Si reaction. Using the $beta$ decay of $^{26}$P to populate the key $J^{pi}=3^+$ resonance in this reaction, we report the first evidence for the observation of its exit channel via a $1741.6 pm 0.6 (textrm{stat}) pm 0.3 (textrm{syst})$ keV primary $gamma$ ray, where the uncertainties are statistical and systematic, respectively. By combining the measured $gamma$-ray energy and intensity with other experimental data on $^{26}$Si, we find the center-of-mass energy and strength of the resonance to be $E_r = 414.9 pm 0.6(textrm{stat}) pm 0.3 (textrm{syst}) pm 0.6(textrm{lit.})$ keV and $omegagamma = 23 pm 6 (textrm{stat})^{+11}_{-10}(textrm{lit.})$ meV, respectively, where the last uncertainties are from adopted literature data. We use hydrodynamic nova simulations to model $^{26}$Al production showing that these measurements effectively eliminate the dominant experimental nuclear-physics uncertainty and we estimate that novae may contribute up to 30% of the Galactic $^{26}$Al.
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