We simulate the dilute attractive Fermi–Hubbard model in the unitarity regime using a diagrammatic determinant Monte Carlo (MC) algorithm with worm-type updates. We obtain the dependence of the critical temperature on the filling factor ν and, by extrapolating to ν → 0, determine the universal critical temperature of the continuum unitary Fermi gas in units of Fermi energy: Tc/εF = 0.152(7). We also determine the thermodynamic functions and show how the MC results can be used for accurate thermometry of a trapped unitary gas.