  We analyse the proton Compton-scattering differential cross section for
photon energies up to 325 MeV using Chiral Effective Field Theory and extract
new values for the electric and magnetic polarisabilities of the proton. Our
EFT treatment builds in the key physics in two different regimes: photon
energies around the pion mass ("low energy") and the higher energies where the
Delta(1232) resonance plays a key role. The Compton amplitude is complete at
N4L0, O(e^2 delta^4), in the low-energy region, and at NLO, O(e^2 delta^0), in
the resonance region. Throughout, the Delta-pole graphs are dressed with pi-N
loops and gamma-N-Delta vertex corrections. A statistically consistent database
of proton Compton experiments is used to constrain the free parameters in our
amplitude: the M1 gamma-N-Delta transition strength b_1 (which is fixed in the
resonance region) and the polarisabilities alpha and beta (which are fixed from
data below 170 MeV). In order to obtain a reasonable fit we find it necessary
to add the spin polarisability gammaM1 as a free parameter, even though it is,
strictly speaking, predicted in chiral EFT at the order to which we work. We
show that the fit is consistent with the Baldin sum rule, and then use that sum
rule to constrain alpha+beta. In this way we obtain
alpha=[10.65+/-0.35(stat})+/-0.2(Baldin)+/-0.3(theory)]10^{-4} fm^3, and beta
=[3.15-/+0.35(stat)-/+0.2(Baldin)-/+0.3(theory)]10^{-4} fm^3, with chi^2 =
113.2 for 135 degrees of freedom. A detailed rationale for the theoretical
uncertainties assigned to this result is provided.
