It has been found in many experiments that the mean square displacement of a Brownian particle x(T) diffusing in a rearranging environment is strictly Fickian, obeying <(x(T))(2)> proportional to T, but the probability distribution function for the displacement is not Gaussian. An explanation of this is that the diffusivity of the particle itself is changing as a function of time. Models for this diffusing diffusivity have been solved analytically in the limit of small time, but simulations were necessary for intermediate and large times. We show that one of the diffusing diffusivity models is equivalent to Brownian motion in the presence of a sink and introduce a class of models for which it is possible to find analytical solutions. Our solution gives <(x(T))(2)> proportional to T for all times and at short times the probability distribution function of the displacement is exponential which crosses over to a Gaussian in the limit of long times and large displacements.