  We present an exact solution for the distribution P(h_m,L) of the maximal
height h_m (measured with respect to the average spatial height) in the steady
state of a fluctuating Edwards-Wilkinson interface in a one dimensional system
of size L with both periodic and free boundary conditions. For the periodic
case, we show that P(h_m,L)=L^{-1/2}f(h_m L^{-1/2}) for all L where the
function f(x) is the Airy distribution function that describes the probability
density of the area under a Brownian excursion over a unit interval. For the
free boundary case, the same scaling holds but the scaling function is
different from that of the periodic case. Numerical simulations are in
excellent agreement with our analytical results. Our results provide an exactly
solvable case for the distribution of extremum of a set of strongly correlated
random variables.
