  We study the spectrum of the Laplacian on the hemisphere with Robin boundary
conditions. It is found that the eigenvalues fall into small clusters around
the Neumann spectrum, and satisfy a Szeg\H{o} type limit theorem. Sharp upper
and lower bounds for the gaps between the Robin and Neumann eigenvalues are
derived, showing in particular that these are unbounded. Further, it is shown
that except for a systematic double multiplicity, there are no multiplicities
in the spectrum as soon as the Robin parameter is positive, unlike the Neumann
case which is highly degenerate. Finally, the limiting spacing distribution of
the desymmetrized spectrum is proved to be the delta function at the origin.
