  Quantum systems coupled to environments exhibit intricate dynamics. The
master equation gives a Markov approximation of the dynamics, allowing for
analytic and numerical treatments. It is ubiquitous in theoretical and applied
quantum sciences. The accuracy of the master equation approximation was so far
proven for small values of the system-environment interaction coupling strength
$\lambda$, under the additional constraint that time $t$ must not exceed an
upper bound, $\lambda^2 t\le$ constant. Here, we show that the Markov
approximation is valid for fixed small coupling strength and for all times. We
also construct a new approximate Markovian dynamics -- a completely positive,
trace preserving semigroup -- which is asymptotically in time exact, to all
orders in the coupling.
