The source of Galactic cosmic rays (with energies up to 10(15) eV) remains unclear, although it is widely believed that they originate in the shock waves of expanding supernova remnants. At present the best way to investigate their acceleration and propagation is by observing the gamma-rays produced when cosmic rays interact with interstellar gas. Here we report observations of an extended region of very-high-energy (> 10(11) eV) gamma-ray emission correlated spatially with a complex of giant molecular clouds in the central 200 parsecs of the Milky Way. The hardness of the gamma-ray spectrum and the conditions in those molecular clouds indicate that the cosmic rays giving rise to the gamma-rays are likely to be protons and nuclei rather than electrons. The energy associated with the cosmic rays could have come from a single supernova explosion around 10(4) years ago.