Extensive molecular dynamics simulations are performed at 6% polydispersity to investigate the phase diagram of a polydisperse hard-sphere fluid and the dynamical behavior in each phase. As the volume fraction 0 is increased, the existence of a supercooled liquid is suggested above phi = 0.5524 by the mean-field analysis proposed recently. The first-order phase transition from supercooled liquid to crystal is then shown to occur around phi = 0.5630. The occurrence of re-entrant melting, a first-order phase transition from crystal to supercooled liquid, is also found around phi = 0.5715. The existence of the glass transition is thus suggested, even in the hard-sphere fluid with small polydispersity.