Fe and Pt are paradigms for ferromagnetism and strong spin-orbit coupling, respectively. Their combination - in an ultrathin Fe film on a Pt(111) substrate - is thus expected to modify the magnetic structures. We report on a theoretical investigation of a monolayer of Fe on Pt(111), using a generalized Heisenberg model that includes the complete spin interaction matrices l_ij computed from first principles. We find a noncollinear periodic configuration that is strongly determined by the Dzyaloshinskii-Moriya interaction. Taking into account a magnetic field to mimic recent experiments, this noncollinear structure solves the disagreement between the experimental magnetization and the average magnetization for a ferromagnetic system. The critical temperature decreases from 670 K to 590 K due to spin-orbit coupling.