A new solution satisfying Richards' equation is derived. The solution, which may be applied for infiltration or capillary rise, is valid for the condition of an arbitrary moisture tension imposed at the soil surface. When written in terms of the moisture tension, the new result is very simple, being derived in terms of a similarity variable. The solution applies when the form of the soil moisture characteristic curve is a particular weighted integral of the gradient of the unsaturated hydraulic conductivity. Thus, if the soil moisture characteristic curve is selected a priori, then this condition determines the hydraulic conductivity. The converse of this statement also applies. The cumulative infiltration derived from the solution is of the form of the Green-Ampt infiltration equation; however, there is no need to assume a steep wetting front as Green and Ampt did. Finally, using the correspondence between Richards' equation and the convection-dispersion equation with a non-linear solute adsorption isotherm, a new exact solution for adsorptive solute transport is derived.