Eynard and Orantin have recently defined invariants of any compact Riemann surface equipped with two meromorphic functions, as a tool for studying enumerative problems in geometry. I will give a brief introduction to these invariants and describe a particular example that encodes the stationary Gromov-Witten invariants of the two-sphere. This brings new insight into the well-studied problem of the Gromov-Witten invariants of the two-sphere. Conversely, we gain insight into the Eynard-Orantin invariants showing that in this example they are related to the Landau-Ginzburg model dual to Gromov-Witten invariants.