  This paper is concerned with the Dirichlet problem for an equation involving
the 1--Laplacian operator $\Delta_1 u$ and having a singular term of the type
$\frac{f(x)}{u^\gamma}$. Here $f\in L^N(\Omega)$ is nonnegative, $0<\gamma\le1$
and $\Omega$ is a bounded domain with Lipschitz--continuous boundary. We prove
an existence result for a concept of solution conveniently defined. The
solution is obtained as limit of solutions of $p$--Laplacian type problems.
Moreover, when $f(x)>0$ a.e., the solution satisfies those features that might
be expected as well as a uniqueness result. We also give explicit
1--dimensional examples that show that, in general, uniqueness does not hold.
We remark that the Anzellotti theory of $L^\infty$--divergence--measure vector
fields must be extended to deal with this equation.
