Measuring Compton scattered photons and recoil neutrons in coincidence, quasifree Compton scattering by the neutron has been investigated at MAMI (Mainz) at theta(lab)(gamma) = 136 degrees in an energy range from 200 to 400 MeV. From the data a polarizability difference of alpha(n)-beta(n) = 9.8+/-3.6(stat)+2.1-1.1(syst)+/-2.2(model) in units of 10(-4) fm(3) has been determined. In combination with the polarizability sum alpha(n)+beta(n) = 15.2+/-0.5 deduced from photoabsorption data, the first precise results for the neutron electric and magnetic polarizabilities, alpha(n) = 12.5+/-1.8(stat)+1.1-0.6(syst)+/-1.1(model) and beta(n) = 2.7-/+1.8(stat)+0.6-1.1(syst)-/+1.1(model), are obtained.