  We prove the modularity of minimally ramified ordinary residually reducible
p-adic Galois representations of an imaginary quadratic field F under certain
assumptions. We first exhibit conditions under which the residual
representation is unique up to isomorphism. Then we prove the existence of
deformations arising from cuspforms on GL_2(A_F) via the Galois representations
constructed by Taylor et al. We establish a sufficient condition (in terms of
the non-existence of certain field extensions which in many cases can be
reduced to a condition on an L-value) for the universal deformation ring to be
a discrete valuation ring and in that case we prove an R=T theorem. We also
study reducible deformations and show that no minimal characteristic 0
reducible deformation exists.
