Gaussian Processes (GPs) are a theoretically grounded mechanism for modelling with both aleatoric and epistemic uncertainty, but, the well-known solutions of the predictive GP posterior distribution apply only for deterministic inputs. If however, the input is uncertain, closed-form solutions aren't generally available and approximation schemes, such as moment-matching and Monte Carlo simulation, must be used. Moment-matching is only available under restricted conditions on the input distribution and the GP prior, and will miss the nuances of the posterior distribution; Monte Carlo simulation can be computationally expensive. In this article, we present a general method that uses a novel uncertainty-tracking microprocessor, Laplace, to implicitly calculate the posterior predictive distribution with uncertain inputs. We show that our method captures the nuances of the posterior distribution while being orders of magnitude faster than Monte Carlo simulation.