  We study the phase diagram of the Hubbard model in the limit where U, the
onsite repulsive interaction, is much smaller than the bandwidth. We present an
asymptotically exact expression for T$_c$, the superconducting transition
temperature, in terms of the correlation functions of the non-interacting
system which is valid for arbitrary densities so long as the interactions are
sufficiently small. Our strategy for computing T$_c$ involves first integrating
out all degrees of freedom having energy higher than an unphysical initial
cutoff $\Omega_0$. Then, the renormalization group (RG) flows of the resulting
effective action are computed and T$_c$ is obtained by determining the scale
below which the RG flows in the Cooper channel diverge. We prove that T$_c$ is
independent of $\Omega_0$. Using this method, we find a variety of
unconventional superconducting ground states in two and three dimensional
lattice systems and present explicit results for T$_c$ and pairing symmetries
as a function of the electron concentration.
