<p>A standard task in solid state physics and quantum chemistry is the computation of localized molecular orbitals known as Wannier functions. In this manuscript, we propose a new procedure for computing Wannier functions in one-dimensional crystalline materials. Our approach proceeds by first performing parallel transport of the Bloch functions using numerical integration. Then, a simple analytically computable correction is introduced to yield the optimally localized Wannier function. The resulting scheme is rapidly convergent and is proven to yield real-valued Wannier functions that achieve global optimality. The analysis in this manuscript can also be viewed as a proof of the existence of exponentially localized Wannier functions in one dimension. We illustrate the performance of the scheme by a number of numerical experiments.</p>