Let $G$ be a $\sigma$-compact, locally compact group and $\mathcal I$ be a closed 2-sided ideal with finite codimension in $L^1(G)$. It isshownthat there are a closed left ideal ${\mathcal L}$having a right boundedapproximate identity anda closed right ideal ${\mathcal R}$ having a leftbounded approximate identity such that${\mathcal I} = {\mathcal L} + {\mathcal R}$.The proof uses ideasfrom the theory ofboundaries of random walks on groups. 2000 Mathematics Subject Classification:primary 43A20; secondary 42A85, 43A07, 46H10, 46H40, 60B11.