we provide concrete (computable) necessary and sufficient conditions for the existence of a representing measure, supported in a prescribed parabola p(x,y) =0, for moment data b := b(2n)={b_{i,j}: i, j, >= 0, i+j <=2n}. there exists a positive borel measure u, supported in p(x,y)=0, such that b_{i,j} is the u-moment for x^i y^j (i+j <= 2n) if and only if the associated moment matrix m(n)(b) is positive semi-definite, recursively generated, has a column dependence relation p(x,y) = 0, and satisfies rank m(n)(b) <= card v(b), where v(b) is the algebraic variety naturally associated to the data b(2n). (joint work with r.e. curto)