Let a₁,...,a_n and b₁,...,b_m be fixed positive integers, and let S denote the set of all nonnegative integer solutions of the equation x₁a₁+...+x_na_n=y₁b₁+...+y_mb_m. A solution (x₁,...,x_n,y₁,...,y_m) in S is called minimal if it cannot be expressed as the sum of two nonzero solutions in S.  For each pair (i,j), with 1 ≤ i ≤ n and 1 ≤ j ≤ m, the solution whose only nonzero coordinates are x_i = b_j and y_j = a_i is called a generator.  We show that every minimal solution is a convex combination of the generators and the zero-solution. This proves a conjecture of Henk-Weismantel and, independently, Hosten-Sturmfels.