We discovered that La-doped BaSnO 3 with the perovskite structure has an unprecedentedly high mobility at room temperature while retaining its optical transparency. In single crystals, the mobility reached 320 cm 2 V -1 s -1 at a doping level of $8\times 10^{19}$ cm -3 , constituting the highest value among wide-band-gap semiconductors. In epitaxial films, the maximum mobility was 70 cm 2 V -1 s -1 at a doping level of $4.4\times 10^{20}$ cm -3 . We also show that resistance of (Ba,La)SnO 3 changes little even after a thermal cycle to 530 °C in air, pointing to an unusual stability of oxygen atoms and great potential for realizing transparent high-frequency, high-power functional devices.