An analytical solution based on Laplace transforms is developed for the problem of radionuclide transport along a discrete planar fracture in porous rock. The solution takes into account advective transport in the fracture, longitudinal hydrodynamic dispersion along the fracture axis, molecular diffusion from the fracture into the rock matrix, sorption within the rock matrix, sorption onto the surface of the fracture, and radioactive decay. The longitudinal-dispersion-free solution, which is a closed form, is also reported. The initial concentration in both the fracture and the rock matrix is assumed to be zero. An exponentially decaying flux is used for the inlet boundary condition. In addition to the radionuclide concentration in both the fracture and the rock matrix, the mass flux in the fracture is provided. The analytical solution is in the form of a single integral that is evaluated by a Gauss-Legendre quadrature for each point in space and time. A comparison between the concentration profiles with a flux-type inlet boundary condition and those with a concentration-type inlet boundary condition shows that the concentration profile is strongly influenced by the inlet boundary condition when the retardation factor of the rock matrix is high. The analytical solution is verified by results generated from a numerical inversion of the Laplace transforms. The agreement is excellent.