Series expansion methods are used to study directed bond percolation clusters on the square lattice whose lateral growth is restricted by a wall parallel to the growth direction. The percolation threshold pc is found to be the same as that for the bulk. However, the values of the critical exponents for the percolation probability and mean cluster size are quite different from those for the bulk and are estimated by β1 = 0.7338 ± 0.0001 and γ1 = 1.8207 ± 0.0004 respectively. On the other hand the exponent Δ1 = β1 + γ1 characterizing the scale of the cluster size distribution is found to be unchanged by the presence of the wall. The parallel connectedness length, which is the scale for the cluster length distribution, has an exponent which we estimate to be ν1∥ = 1.7337±0.0004 and is also unchanged. The exponent τ1 of the mean cluster length is related to β1 and ν1∥ by the scaling relation ν1∥ = β1 + τ1 and using the above estimates yields τ1 = 1 to within the accuracy of our results. We conjecture that this value of τ1 is exact and further support for the conjecture is provided by the direct series expansion estimate τ1 = 1.0002 ± 0.0003.