  We build a finite volume scheme for the scalar conservation law $\partial_t u
+ \partial_x (H(x, u)) = 0$ with bounded initial condition for a wide class of
flux function $H$, convex with respect to the second variable. The main idea
for the construction of the scheme is to use the theory of discontinuous flux.
We prove that the resulting approximating sequence converges boundedly almost
everywhere on $\mathopen]0, +\infty\mathclose[$ to the entropy solution.
