One of the embarrassments of covariant string field theory has been the glaring failure to derive the Shapiro-Virasoro amplitude. Modular invariance appears explicitly violated: either the fundamental region is overcounted an infinite number of times, or it is undercounted because of a missing region. We try to approach this problem from a fresh point of view. Conventional wisdom holds that, in string field theory, the Veneziano amplitude can only be derived either in light-cone string field theory or Witten's string field theory. We show that this firmly held belief is actually wrong, that the Veneziano amplitude can actually be derived using vertices of {ital arbitrary} lengths. This is a highly nontrivial calculation. Using third elliptic integrals we show that a series of miracles'' occurs which allow us to cancel scores of unwanted terms in the measure, leaving us with the correct Koba-Nielsen variable and measure. We give three independent proofs of our result. When we generalize our results to closed-string scattering with arbitrary lengths, we find a new surprise, that we can successfully derive the Shapiro-Virasoro amplitude as long as a crucial four-string interaction term is added. We check by explicit computer calculation that we reproduce the correct region ofmore » integration for the four-closed-string amplitude. Crucial to the theory is the existence of the missing four-string tetrahedron graph, which precisely fills the missing integration region. We comment on the implications of this for geometric string field theory.« less