  We identify the zero dispersion limit of a solution of the Benjamin--Ono
equation on the line corresponding to every initial datum in $L^2(\R)\cap
L^\infty(\R )$. We infer a maximum principle and a local smoothing property for
this limit. The proof is based on an explicit formula for the Benjamin--Ono
equation and on the combination of calculations in the special case of rational
initial data, with approximation arguments. We also investigate the special
case of an initial datum equal to the characteristic function of a finite
interval, and prove the lack of semigroup property for this zero dispersion
limit.
