  We show that the Veneziano amplitude of string theory is the unique solution
to an analytically solvable bootstrap problem. Uniqueness follows from two
assumptions: faster than power-law falloff in high-energy scattering and the
existence of some infinite sequence in momentum transfer at which higher-spin
exchanges cancel. The string amplitude$\unicode{x2013}$including the mass
spectrum$\unicode{x2013}$is an output of this bootstrap. If the amplitude
merely vanishes at high energies, the solution is a three-parameter family
containing the Veneziano, Coon, and hypergeometric amplitudes, and more.
