A precise measurement of the differential cross sections dσ/dΩ and the linearly polarized photon beam asymmetry Σ₃ for Compton scattering on the proton below pion threshold has been performed with a tagged photon beam and almost 4π 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 setup. 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: α_E1 = 10.99 ± 0.16 ± 0.47 ± 0.17 ± 0.34, β_M1 = 3.14 ± 0.21 ± 0.24 ± 0.20 ± 0.35; in units of 10^-4 fm³ where the errors are statistical, systematic, spin polarizability dependent, and model dependent.