We prepared one-unit-cell (1-UC) thick FeSe films on insulating SrTiO3 substrates with non-superconducting FeTe protection layers by molecular beam epitaxy for ex situ studies. By direct transport and magnetic measurements, we provide definitive evidence for high temperature superconductivity in the 1-UC FeSe films with an onset TC above 40 K and an extremely large critical current density JC∼1.7×106 A/cm2 at 2 K, which are much higher than TC∼8 K and JC∼104 A/cm2 for bulk FeSe, respectively. Our work may pave the way to enhancing and tailoring superconductivity by interface engineering.