  A precise measurement of the differential cross-sections $d\sigma/d\Omega$
and the linearly polarized photon beam asymmetry $\Sigma_3$ for Compton
scattering on the proton below pion threshold has been performed with a tagged
photon beam and almost $4\pi$ detector at the Mainz Microtron. The incident
photons were produced by the recently upgraded Glasgow-Mainz photon tagging
facility and impinged on a cryogenic liquid hydrogen target, with the scattered
photons detected in the Crystal Ball/TAPS set-up. Using the highest statistics
Compton scattering data ever measured on the proton along with two effective
field theories (both covariant baryon and heavy-baryon) and one fixed-$t$
dispersion relation model, constraining the fits with the Baldin sum rule, we
have obtained the proton electric and magnetic polarizabilities with
unprecedented precision: \begin{align*}
  &{}\alpha_{E1} = 10.99 \pm 0.16 \pm 0.47 \pm 0.17 \pm 0.34
  &{}\beta_{M1} = 3.14 \pm 0.21 \pm 0.24 \pm 0.20 \pm 0.35 \end{align*} in
units of $10^{-4}$\,fm$^3$ where the errors are statistical, systematic, spin
polarizability dependent and model dependent.
