  We compute the universal deformations of cuspidal representations $\pi$ of
$\GL_2(F)$ over an algebraically closed field of characteristic $l$, where $F$
is a local field of residue characteristic $p$ not equal to $l$. When $\pi$ is
supercuspidal there is an irreducible, two-dimensional representation $\rho$ of
$G_F$ that corresponds to $\pi$ by the mod $l$ local Langlands correspondence
of Vign{\'e}ras; we show there is a natural isomorphism between the universal
deformation rings of $\pi$ and $\rho$ that induces the usual local Langlands
correspondence on characteristic zero points. Our work establishes certain
cases of a conjecture of Emerton.
