Quantum dots are often called artificial atoms because, like real atoms, they confine electrons to quantized states with discrete energies. However, although real atoms are identical, most quantum dots comprise hundreds or thousands of atoms, with inevitable variations in size and shape and, consequently, unavoidable variability in their wavefunctions and energies. Electrostatic gates can be used to mitigate these variations by adjusting the electron energy levels(1-3), but the more ambitious goal of creating quantum dots with intrinsically digital fidelity by eliminating statistical variations in their size, shape and arrangement remains elusive(4-9). We used a scanning tunnelling microscope to create quantum dots with identical, deterministic sizes. By using the lattice of a reconstructed semiconductor surface to fix the position of each atom, we controlled the shape and location of the dots with effectively zero error. This allowed us to construct quantum dot molecules whose coupling has no intrinsic variation but could nonetheless be tuned with arbitrary precision over a wide range. Digital fidelity opens the door to quantum dot architectures free of intrinsic broadening-an important goal for technologies from nanophotonics(10) to quantum information processing(11,12) as well as for fundamental studies of confined electrons(13-17).