  We present a completely explicit transcendence measure for $e$. This is a
continuation and an improvement to the works of Borel, Mahler and Hata on the
topic. Furthermore, we also prove a transcendence measure for an arbitrary
positive integer power of $e$. The results are based on Hermite-Pad\'e
approximations and on careful analysis of common factors in the footsteps of
Hata.
