We report results of a study of the ${B}_{s}^{0}$ oscillation frequency using a large sample of ${B}_{s}^{0}$ semileptonic decays corresponding to approximately $1\text{ }\text{ }{\mathrm{fb}}^{\ensuremath{-}1}$ of integrated luminosity collected by the D0 experiment at the Fermilab Tevatron Collider in 2002--2006. The amplitude method gives a lower limit on the ${B}_{s}^{0}$ oscillation frequency at $14.8\text{ }\text{ }{\mathrm{ps}}^{\ensuremath{-}1}$ at the 95% C.L. At $\ensuremath{\Delta}{m}_{s}=19\text{ }\text{ }{\mathrm{ps}}^{\ensuremath{-}1}$, the amplitude deviates from the hypothesis $\mathcal{A}=0$ (1) by 2.5 (1.6) standard deviations, corresponding to a two-sided C.L. of 1% (10%). A likelihood scan over the oscillation frequency, $\ensuremath{\Delta}{m}_{s}$, gives a most probable value of $19\text{ }\text{ }{\mathrm{ps}}^{\ensuremath{-}1}$ and a range of $17<\ensuremath{\Delta}{m}_{s}<21\text{ }\text{ }{\mathrm{ps}}^{\ensuremath{-}1}$ at the 90% C.L., assuming Gaussian uncertainties. This is the first direct two-sided bound measured by a single experiment. If $\ensuremath{\Delta}{m}_{s}$ lies above $22\text{ }\text{ }{\mathrm{ps}}^{\ensuremath{-}1}$, then the probability that it would produce a likelihood minimum similar to the one observed in the interval $16--22\text{ }\text{ }{\mathrm{ps}}^{\ensuremath{-}1}$ is $(5.0\ifmmode\pm\else\textpm\fi{}0.3)%$.