Thermodynamically rigorous free energy methods in principle allow the exact computation of binding free energies in biological systems. Here, we use thermodynamic integration together with molecular dynamics simulations of a DNA-protein complex to compute relative binding free energies of a series of mutants of a protein-binding DNA operator sequence. A guanine-cytosine basepair that interacts strongly with the DNA-binding protein is mutated into adenine-thymine, cytosine-guanine, and thymine-adenine. It is shown that basepair mutations can be performed using a conservative protocol that gives error estimates of 10% of the change in free energy of binding. Despite the high CPU-time requirements, this work opens the exciting opportunity of being able to perform basepair scans to investigate protein-DNA binding specificity in great detail computationally.