We show that Monte Carlo sampling of the Feynman diagrammatic series (DiagMC) can be used for tackling hard fermionic quantum many-body problems in the thermodynamic limit by presenting accurate results for the repulsive Hubbard model in the correlated Fermi liquid regime. Sampling Feynman's diagrammatic series for the single-particle self-energy, we can study moderate values of the on-site repulsion (U/t~4) and temperatures down to T/t=1/40. We compare our results with high-temperature series expansions and with single-site and cluster dynamical mean-field theory.