We present a measurement of the $t\overline{t}$ production cross section in $p\overline{p}$ collisions at $\sqrt{s}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1.8\mathrm{TeV}$ by the D0 experiment at the Fermilab Tevatron. The measurement is based on data from an integrated luminosity of approximately $125{\mathrm{pb}}^{\ensuremath{-}1}$ accumulated during the 1992--1996 collider run. We observe 39 $t\overline{t}$ candidate events in the dilepton and $\mathrm{lepton}+\mathrm{jets}$ decay channels with an expected background of $13.7\ifmmode\pm\else\textpm\fi{}2.2$ events. For a top quark mass of $173.3\mathrm{GeV}{/c}^{2}$, we measure the $t\overline{t}$ production cross section to be $5.5\ifmmode\pm\else\textpm\fi{}1.8\mathrm{pb}$.