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 is shown that there are a closed left ideal ${\mathcal L}$ having a right bounded approximate identity and a closed right ideal ${\mathcal R}$ having a left bounded approximate identity such that ${\mathcal I} = {\mathcal L} + {\mathcal R}$. The proof uses ideas from the theory of boundaries of random walks on groups. 2000 Mathematics Subject Classification: primary 43A20; secondary 42A85, 43A07, 46H10, 46H40, 60B11.