We report angle-resolved photoemission measurements of the dispersion of the shallow surface state around (A)overbar and the surface electron-phonon mass-enhancement parameter lambda(s) at the Be(1010) surface. The eccentricity of the elliptical Fermi line formed by the surface state is found to be epsilon=0.684. We obtain lambda(s)=0.672 +/- 0.027 along the major axis and lambda(s)= 0.642+/-0.031 along the minor axis of the Fermi line. The lambda(s) values are about three times larger than the bulk lambda(b) = 0.24.