Single-layer transition metal dichalcogenide WSe2 has recently attracted a lot of attention because it is a 2D semiconductor with a direct band gap. Due to low doping levels, it is intrinsic and shows ambipolar transport. This opens up the possibility to realize devices with the Fermi level located in the valence band, where the spin/valley coupling is strong and leads to new and Interesting physics. As a consequence of its intrinsically low doping, large Schottky barriers form between WSe2 and metal contacts, which impede the injection of charges at low temperatures. Here, we report on the study of single-layer WSe2 transistors with a polymer electrolyte gate (PEO:LiCIO4). Polymer electrolytes allow the charge carrier densities to be modulated to very high values, allowing the observation of both the electron- and the hole-doped regimes. Moreover, our ohmic contacts formed at low temperatures allow us to study the temperature dependence of electron and hole mobilities. At high electron densities, a re-entrant insulating regime is also observed, a feature which is absent at high hole densities.