The adsorption of S on the $\mathrm{Si}(001)c(4\ifmmode\times\else\texttimes\fi{}2)$ surface is studied by first-principles total-energy calculations. We started with the adsorption of a single atom up to a full-monolayer coverage. The first S atom occupies a bridge site, on top of a Si dimer that becomes completely symmetric. At half-monolayer coverage, all bridge sites are occupied, and all Si dimers become symmetric. The overall periodicity is $(2\ifmmode\times\else\texttimes\fi{}1).$ The adsorption of an additional S atom results in the breaking of two Si dimers. From this point and up to one monolayer, it is energetically more favorable for the S atoms to be adsorbed along the $[1\ifmmode\bar\else\textasciimacron\fi{}10]$ direction. At full monolayer coverage, all Si dimers are broken, and the Si surface is dereconstructed with all Si atoms near bulk ideal positions.