Unique decomposition for a polynomial of low rank

Let F be a homogeneous polynomial of degree d in m+1 variables defined over an algebraically closed field of characteristic 0 and suppose that F belongs to the sth secant variety of the d-uple Veronese embedding of P-m into P((m+d)(d))(-1) but that its minimal decomposition as a sum of dth powers of linear forms requires more than s summands. We show that if s <= d then F can be uniquely written as F = M-1(d) + ... + M-t(d) + Q, where M-1, ... , M-t are linear forms with t <= (d - 1)/2, and Q is a binary form such that Q = Sigma(q)(i=1) l(i)(d-di)m(i) with l(i)'s linear forms and m(i)'s forms of degree d(i) such that Sigma(d(i) + 1) = s - t.