  We obtain a complete and exact in the weak-coupling limit ($U \rightarrow 0$)
ground state phase diagram of the repulsive fermionic Hubbard model on the
square lattice for filling factors $0 < n < 2$ and next-nearest-neighbour
hopping amplitudes $0 \le t^{\prime} \le 0.5$. Phases are distinguished by the
symmetry and the number of nodes of the superfluid order parameter. The phase
diagram is richer than may be expected and typically features states with a
high --- higher than that of the fundamental mode of the corresponding
irreducible representation --- number of nodes. The effective coupling strength
in the Cooper channel $\lambda$, which determines the critical temperature
$T_c$ of the superfluid transition, is calculated in the whole parameter space
and regions with high values of $\lambda$ are identified. It is shown that
besides the expected increase of $\lambda$ near the Van Hove singularity line,
joining the ferromagnetic and antiferromagnetic points, another region with
high values of $\lambda$ can be found at quarter filling and $t^{\prime}=0.5$
due to the presence of a line of nesting at $t^{\prime} \ge 0.5$. The results
can serve as benchmarks for controlled non-perturbative methods and guide the
ongoing search for high-$T_c$ superconductivity in the Hubbard model.
