  Given a directed graph $E$ and a labeling $\mathcal{L}$, one forms the
labeled graph $C^*$-algebra by taking a weakly left--resolving labeled space
$(E, \mathcal{L}, \mathcal{B})$ and considering a universal generating family
of partial isometries and projections. In this paper, we work on ideals for a
labeled graph $C^*$-algebra when the graph contains sinks. Using some of the
tools we build, we compute $C^*(E, \mathcal{L}, \mathcal{B})$ when $E$ is a
finite graph.
