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)