The absolute branching fractions for the decays $D^-_s\rightarrow\ell^-\bar{\nu}_{\ell}$ ($\ell=e$, $\mu$, or $\tau$) are measured using a data sample corresponding to an integrated luminosity of 521 fb$^{-1}$ collected at center of mass energies near 10.58 GeV with the BABAR detector at the PEP-II $e^+e^-$ collider at SLAC. The number of $D_s^-$ mesons is determined by reconstructing the recoiling system $DKX\gamma$ in events of the type $e^+e^-{\rightarrow}DKXD^{*-}_s$, where $D^{*-}_s\rightarrow D^-_s\gamma$ and $X$ represents additional pions from fragmentation. The $D^-_s\rightarrow\ell^-\nu_{\ell}$ events are detected by full or partial reconstruction of the recoiling system $DKX\gamma\ell$. The following results are obtained: $B(D^-_s \to \mu^- \nu) = (6.02 \pm 0.38 \pm 0.34) \times 10^{-3}$, $B(D^-_s \to \tau^- \nu) = (5.00 \pm 0.35 \pm 0.49) \times 10^{-2}$, and $B(D^-_s \to e^- \nu)