  We present the exact Bethe ansatz solution for the two-dimensional BCS
pairing Hamiltonian with p_x + i p_y symmetry. Using both mean-field theory and
the exact solution we obtain the ground-state phase diagram parameterized by
the filling fraction and the coupling constant. It consists of three phases
denoted weak coupling BCS, weak pairing, and strong pairing. The first two
phases are separated by a topologically protected line where the exact ground
state is given by the Moore-Read pfaffian state. In the thermodynamic limit the
ground-state energy is discontinuous on this line. The other two phases are
separated by the critical line, also topologically protected, previously found
by Read and Green. We establish a duality relation between the weak and strong
pairing phases, whereby ground states of the weak phase are "dressed" versions
of the ground states of the strong phase by zero energy (Moore-Read) pairs and
characterized by a topological order parameter.
