An approximation to the transport equation is presented, which is capable of arbitrary accuracy and yields the exact transport-theory asymptotic behavior in all orders for any geometry. Anisotropic scattering is treated explicitly, and the inclusion of energy and time dependences is straightforward. The approximation, which is very similar to the usual spherical-harmonic (PN) method, is derived by introducing a new truncation scheme into the infinite set spherical-harmonic equations. This truncation method consists of assuming that the higher spherical-harmonic components, equated to zero in the PN method, can be related to lower components by assuming the angular distribution to be in an asymptotic distribution. The resulting approximation is very similar in structure to the PN approximation (in particular, it is no more complex) but has the added advantage of yielding exact asymptotic behavior.