  $^{13}$N($p,\gamma$)$^{14}$O is one of the key reactions in the hot CNO cycle
which occurs at stellar temperatures around $T_9$ $\geq$ 0.1. Up to now, some
uncertainties still exist for the direct capture component in this reaction,
thus an independent measurement is of importance. In present work, the angular
distribution of the $^{13}$N($d,n$)$^{14}$O reaction at $E_{\rm{c.m.}}$ = 8.9
MeV has been measured in inverse kinematics, for the first time. Based on the
distorted wave Born approximation (DWBA) analysis, the nuclear asymptotic
normalization coefficient (ANC), $C^{^{14}O}_{1,1/2}$, for the ground state of
$^{14}$O $\to$ $^{13}$N + $p$ is derived to be $5.42 \pm 0.48$ fm$^{-1/2}$. The
$^{13}$N($p,\gamma$)$^{14}$O reaction is analyzed with the R-matrix approach,
its astrophysical S-factors and reaction rates at energies of astrophysical
relevance are then determined with the ANC. The implications of the present
reaction rates on the evolution of novae are then discussed with the reaction
network calculations.
