A significant part of the T=0 phase diagram of the two-dimensional Fermi-Hubbard
model (U<5, n <0.8) is controlled by Fermi liquid physics with weak effective
BCS-type coupling. We access this regime in a controlled way using bold-line
diagrammatic Monte Carlo technique in combination with the semi-analytic treatment
of the weak instability in the Cooper channel. We obtain the corresponding T=0 phase diagram
in the (n,U) plane, describing the competition between the p- and d-wave superfluid states.
We also claim the values of the dimensionless BCS coupling constants---controlling the superfluid Tc---at
the phase boundaries, which prove to be very small up to U=4, n = 0.6.