We start with the familiar Watson-Sommerfeld formula for an amplitude with an infinite sequence of parallel Regge trajectories and an arbitrary set of physical residues. This input is supplemented by assuming two features of all Veneziano models, namely, the narrow-resonance approximation and the absence of cuts in the Mandelstam variables. With this input we show that the scattering amplitude can always be expressed as an infinite series of Veneziano-like terms. The coefficients of this series are uniquely and explicitly expressed in terms of the residues of the original Regge poles with which we started. Each coefficient is expressed in terms of a power of the forward difference operator acting on a well-defined, though complicated, linear combination of the Regge residues. The convergence of this series is studied in detail, and sufficient conditions are states that guarantee absolute convergence away from the poles. These conditions are stated in terms of bounds on the growth in s and in the sequence index n of certain linear combinations of the Regge residues. This is the only input we require in addition to the usual Regge input supplemented by the narrow-resonance approximation and the absence of branch cuts. Duality is not taken as an input; however, we only deal with a fully crossing-symmetric case. The fact that we start with the physical Regge formula guarantees that our Veneziano series has no ghosts as long as the β's we use to define the coefficients are physical. A direct check of this fact is given. Having an explicit expression for the coefficients of the series, we also investigate the conditions that will lead to dominance by the primary term. These turn out to give a form for the leading residue which is almost identical to the one that is used in practical fits to the high-energy data. In addition, it turns out that one has to choose the scale that determines the onset of Regge behavior, s0, to be equal to the inverse of the slope of the Regge trajectory. This again is close to the value of s0 used in practice.