Let H:=−12Δ+V be a one-dimensional continuum Schrödinger operator. Consider Hˆ:=H+ξ, where ξ is a translation invariant Gaussian noise. Under some assumptions on ξ, we prove that if V is locally integrable, bounded below, and grows faster than log at infinity, then the semigroup e−tHˆ is trace class and admits a probabilistic representation via a Feynman-Kac formula. Our result applies to operators acting on the whole line R, the half line (0,∞), or a bounded interval (0,b), with a variety of boundary conditions. Our method of proof consists of a comprehensive generalization of techniques recently developed in the random matrix theory literature to tackle this problem in the special case where Hˆ is the stochastic Airy operator.