The ratio of the proton's elastic electromagnetic form factors, G(E(p))/G(M(p)), was obtained by measuring P(t) and P(l), the transverse and the longitudinal recoil proton polarization, respectively. For elastic e-->p-->ep-->, G(E(p))/G(M(p)) is proportional to P(t)/P(l). Simultaneous measurement of P(t) and P(l) in a polarimeter provides good control of the systematic uncertainty. The results for the ratio G(E(p))/G(M(p)) show a systematic decrease as Q2 increases from 0.5 to 3.5 GeV2, indicating for the first time a definite difference in the spatial distribution of charge and magnetization currents in the proton.