  We prove modularity of certain residually reducible ordinary 2-dimensional
$p$-adic Galois representations with determinant a finite order odd character
$\chi$. For certain non-quadratic $\chi$ we prove an $R=T$ result for $T$ the
weight 1 specialisation of the Hida Hecke algebra acting on non-classical
weight 1 forms, under the assumption that no two Hida families congruent to an
Eisenstein series cross in weight 1. For quadratic $\chi$ we prove that the
quotient of $R$ corresponding to deformations split at $p$ is isomorphic to the
Hecke algebra acting on classical CM weight 1 modular forms.
