We report on laser cooling of a large fraction of positronium (Ps) in free flight by strongly saturating the 1^{3}S-2^{3}P transition with a broadband, long-pulsed 243 nm alexandrite laser. The ground state Ps cloud is produced in a magnetic and electric field-free environment. We observe two different laser-induced effects. The first effect is an increase in the number of atoms in the ground state after the time Ps has spent in the long-lived 2^{3}P states. The second effect is one-dimensional Doppler cooling of Ps, reducing the cloud's temperature from 380(20) to 170(20) K. We demonstrate a 58(9)% increase in the fraction of Ps atoms with v_{1D}<3.7×10^{4} ms^{-1}.