The best constant C-n,C-m in the Jackson-Nikol'skii inequality between the uniform and integral norms of algebraic polynomials of a given total degree n >= 0 on the unit sphere Sm-1 of the Euclidean space R-m is studied. Two-sided estimates for the constant Cn, m are obtained, which, in particular, give the order n(m-1) of its behavior with respect to n as n -> +infinity for a fixed m.