The well known asymptotic diffusion approximation was first developed in the 50's by Frankel and Nelson, and expanded by Case et al. and by Davison, to handle the asymptotic steady-state behavior. But, in time-dependent problems, the parabolic nature of the diffusion equation predicts that particles will have an infinite velocity; particles at the tail of the distribution function will show up at infinite distance from a source in time t = 0+. The classical P 1 approximation (or the equivalent Telegrapher's equation) has a finite particle velocity, but with the wrong value, namely υ/√3. In this work we develop a new approximation from the asymptotic solution of the time-dependent Boltzmann equation, which includes the correct eigenvalue of the asymptotic diffusion approximation and the (almost) correct time behavior (such as the particle velocity), for a general medium. The resulting scalar flux from the new approximation shows a good agreement with the exact solution of the Boltzmann equation.