  We consider a last-passage directed percolation model in $Z_+^2$, with i.i.d.
weights whose common distribution has a finite $(2+p)$th moment. We study the
fluctuations of the passage time from the origin to the point
$\big(n,n^{\lfloor a \rfloor}\big)$. We show that, for suitable $a$ (depending
on $p$), this quantity, appropriately scaled, converges in distribution as
$n\to\infty$ to the Tracy-Widom distribution, irrespective of the underlying
weight distribution. The argument uses a coupling to a Brownian directed
percolation problem and the strong approximation of Koml\'os, Major and
Tusn\'ady.
