  Given a reducible Galois representation $\overline{\rho}: G_{\mathbb{Q}}
\rightarrow GL_2( \mathbb{F}_q)$ we show there exists an irreducible
deformation $\rho : G_{\mathbb{Q}} \rightarrow GL_2 (\mathbb{W} [[T_1, T_2,..,
T_r,....,]])$ of $\overline{\rho}$ ramified at infinitely many primes, where
$\mathbb{W}$ denotes the ring of Witt vectors of $\mathbb{F}_q$. This is a
modification of Ramakrishna's result for the irreducible case.
