  This paper concerns the inverse spectral problem for analytic simple surfaces
of revolution. By `simple' is meant that there is precisely one critical
distance from the axis of revolution. Such surfaces have completely integrable
geodesic flows with global action-angle variables and possess global quantum
Birkhoff normal forms (Colin de Verdiere). We prove that isospectral surfaces
within this class are isometric. The first main step is to show that the normal
form at meridian geodesics is a spectral invariant. The second main step is to
show that the metric is determined from this normal form.
