  As a sequel to our proof of the analog of Serre's conjecture for function
fields in Part I of this work, we study in this paper the deformation rings of
$n$-dimensional mod $\ell$ representations $\rho$ of the arithmetic fundamental
group $\pi_1(X)$ where $X$ is a geometrically irreducible, smooth curve over a
finite field $k$ of characteristic $p$ ($\neq \ell$). We are able to show in
many cases that the resulting rings are finite flat over $\BZ_\ell$. The proof
principally uses a lifting result of the authors in Part I of this two-part
work, Taylor-Wiles systems and the result of Lafforgue. This implies a
conjecture of A.J. ~de Jong for representations with coefficients in power
series rings over finite fields of characteristic $\ell$, that have this mod
$\ell$ representation as their reduction.
