The recent discovery of possible high-temperature superconductivity in single-layer FeSe films(1,2) has generated significant experimental and theoretical interest(3,4). In both the cuprate(5,6) and the iron-based(7-11) high-temperature superconductors, superconductivity is induced by doping charge carriers into the parent compound to suppress the antiferromagnetic state. It is therefore important to establish whether the superconductivity observed in the single-layer sheets of FeSe-the essential building blocks of the Fe-based superconductors-is realized by undergoing a similar transition. Here we report the phase diagram for an FeSe monolayer grown on a SrTiO3 substrate, by tuning the charge carrier concentration over a wide range through an extensive annealing procedure. We identify two distinct phases that compete during the annealing process: the electronic structure of the phase at low doping (N phase) bears a clear resemblance to the antiferromagnetic parent compound of the Fe-based superconductors, whereas the superconducting phase (S phase) emerges with the increase in doping and the suppression of the N phase. By optimizing the carrier concentration, we observe strong indications of superconductivity with a transition temperature of 65 +/- 5 K. The wide tunability of the system across different phases makes the FeSe monolayer ideal for investigating not only the physics of superconductivity, but also for studying novel quantum phenomena more generally.