Summary: We describe a systematic approach to the recovery of a function analytic in the upper half-plane, \(C^+\), from measurements over a finite interval on the real axis, \(D\subset R\). Analytic continuation problems of this type are well known to be ill-posed. Thus, the best one can hope for is a simple, linear procedure which exposes this underlying difficulty and solves the problem in a least-squares sense. To accomplish this, we first construct an explicit analytic approximation of the desired function and recast the continuation problem in terms of a `residual function' defined on the measurement window \(D\) itself. The resulting procedure is robust in the presence of noise, and we demonstrate its performance with some numerical experiments.