Planets occur most frequently around cool dwarfs, but only a handful of specific examples are known to orbit the latest-type M stars. Using TESS photometry, we report the discovery of two planets transiting the low-mass star called LP 791-18 (identified by TESS as TOI 736). This star has spectral type M6V, effective temperature 2960 K, and radius 0.17 R o, making it the third-coolest star known to host planets. The two planets straddle the radius gap seen for smaller exoplanets; they include a 1.1R ⊕ planet on a 0.95 day orbit and a 2.3R ⊕ planet on a 5 day orbit. Because the host star is small the decrease in light during these planets' transits is fairly large (0.4% and 1.7%). This has allowed us to detect both planets' transits from ground-based photometry, refining their radii and orbital ephemerides. In the future, radial velocity observations and transmission spectroscopy can both probe these planets' bulk interior and atmospheric compositions, and additional photometric monitoring would be sensitive to even smaller transiting planets.