  The Compton scattering cross section on the proton has been measured at
laboratory angles of 90$^\circ$ and 135$^\circ$ using tagged photons in the
energy range 70--100 MeV and simultaneously using untagged photons in the range
100--148~MeV. With the aid of dispersion relations, these cross sections were
used to extract the electric and magnetic polarizabilities, $\bar{\alpha}$ and
$\bar{\beta}$ respectively, of the proton. We find
  $$\bar{\alpha}+\bar{\beta} =
  ( 15.0 \pm 2.9 \pm 1.1 \pm 0.4 ) \times 10^{-4} \: {\rm fm}^3,$$
  in agreement with a model-independent dispersion sum rule, and
  $$\bar{\alpha}-\bar{\beta} =
  ( 10.8 \pm 1.1 \pm 1.4 \pm 1.0 ) \times 10^{-4} \: {\rm fm}^3,$$
  where the errors shown are statistical, systematic, and model-dependent,
respectively. A comparison with previous experiments is given and global values
for the polarizabilities are extracted.
