The beta-delayed neutron branching ratios (P{betan}) for nuclei near doubly magic 78Ni have been directly measured using a new method combining high-resolution mass separation, reacceleration, and digital beta-gamma spectroscopy of 238U fission products. The P{betan} values for the very neutron-rich isotopes ;{76-78}Cu and 83Ga were found to be much higher than previously reported and predicted. Revised calculations of the betan process, accounting for new mass measurements and an inversion of the pi2p{3/2} and pi1f{5/2} orbitals, are in better agreement with these new experimental results.