The bilinear maximal operator defined below maps $L^p imes L^q$ into $L^r$ provided $10}frac1{2t}int_{-t}^tabs{f(x+y)g(x-y)} dy.$$ In particular $Mfg$ is integrable hinspace if $f$ and $g$ are square integrable, answering a conjecture posed by Alberto Calderón.