This paper employs the unified transform, also known as the Fokas method, to solve the advection-dispersion equation on the half-line. This method combines complex analysis with numerics. Compared to classical approaches used to solve linear partial differential equations (PDEs), the unified transform avoids the solution of ordinary differential equations and, more importantly, constructs an integral representation of the solution in the complex plane which is uniformly convergent at the boundaries. As a consequence, such solutions are well suited for numerical computations. Indeed, the numerical evaluation of the solution requires only the computation of a single contour integral involving an integrand which decays exponentially fast for large values of the integration variable. A novel contribution of this paper, with respect to the solution of linear evolution PDEs in general, and the implementation of the unified transform in particular, is the following: using the advection-dispersion equation as a generic example, it is shown that if the transforms of the given data can be computed analytically, then the unified transform yields a fast and accurate method that converges exponentially with the number of evaluations N yet only has complexity O(N). Furthermore, if the transforms are computed numerically using M evaluations, the unified transform gives rise to a method with complexity O(NM). Results are successfully compared to other existing solutions.