New measurements have been made of the proton and deuteron spin structure functions,gp1andgd1, using deep‐inelastic scattering of longitudinally polarized electrons from polarized NH3and ND3targets. The data span the kinematic range 0.029<x<0.8 andQ2<10 (GeV/c)2. The integrals &Ggr;p=∫10gp1(x,Q2)dxand &Ggr;d=∫10gd1(x,Q2)dxhave been evaluated at fixedQ2=3 (GeV/c)2to yield &Ggr;p=0.127±0.004(stat.)±0.010(syst.) and &Ggr;d=0.041±0.003±0.004. Both measurements are well‐below the Ellis‐Jaffe predictions.