It is a theorem of Siegel that the Weierstrass model y^2 = x^3 + Ax + B of an elliptic curve has finitely many integral points. A “random” such curve should have no points at all. I will show that the average number of integral points on such curves (ordered by height) is bounded — in fact, by 67. The methods combine a Mumford-type gap principle, LP bounds in sphere packing, and results in Diophantine approximation. The same result also holds (though I have not computed an explicit constant) for the families y^2 = x^3 + Ax, y^2 = x^3 + B, and y^2 = x^3 – n^2 x.