  We consider partially directed walks crossing a $L\times L$ square weighted
according to their length by a fugacity $t$. The exact solution of this model
is computed in three different ways, depending on whether $t$ is less than,
equal to or greater than 1. In all cases a complete expression for the dominant
asymptotic behaviour of the partition function is calculated. The model admits
a dilute to dense phase transition, where for $0 < t < 1$ the partition
function scales exponentially in $L$ whereas for $t>1$ the partition function
scales exponentially in $L^2$, and when $t=1$ there is an intermediate scaling
which is exponential in $L \log{L}$.
