The author considers directed percolation on the square lattice, with probability pH(pV) for the horizontal (vertical) bonds to be unbroken. For pH=1- epsilon ( epsilon small) and pV> sigma ( epsilon ) the percolating cluster is asymptotically within a cone phi +< phi < phi +. The author calculates phi +or- and the fluctuations of the boundaries at phi = phi +or- as power series in epsilon , up to terms approximately epsilon 2, showing that the transverse spread of the percolating cluster is random walk-like. For any given pH the percolation threshold pV,c( epsilon ), defined by phi += phi - at pV=pV,c is also calculated.