In 1981, Dennis and Walker developed a convergence theory for structured secant methods which included the PSB and the DFP secant methods, but not the straightforward structured version of the BFGS secant method. Here we fill this gap in the theory by establishing a convergence theory for the structured BFGS secant method. A direct application of our new theory gives the first proof of local and q-superlinear convergence of the important structured BFGS secant method for the nonlinear least-squares problem which is used by Dennis, Gay and Welsh in the current version of the popular and successful NL2SOL code.