We present nanofabrication and mechanical measurements of single-crystal diamond cantilevers with thickness down to 85 nm, thickness uniformity better than 20 nm, and lateral dimensions up to 240 um. Quality factors exceeding one million are found at room temperature, surpassing those of state-of-the-art single-crystal silicon cantilevers of similar dimensions by roughly an order of magnitude. Force sensitivities of a few hundred zeptonewtons result for the best cantilevers at millikelvin temperatures. Single-crystal diamond could thus directly improve existing force and mass sensors by a simple substitution of resonator material, and lead to quantum nanomechanical devices with exceptionally low energy dissipation.