Let X be a del Pezzo surface of degree one over an algebraically closed field
(of any characteristic), and let Cox(X) be its total coordinate ring. We prove
the missing case of a conjecture of Batyrev and Popov, which states that Cox(X)
is a quadratic algebra. We use a complex of vector spaces whose homology
determines part of the structure of the minimal free Pic(X)-graded resolution
of Cox(X) over a polynomial ring. We show that sufficiently many Betti numbers
of this minimal free resolution vanish to establish the conjecture.