Check your signs - the formula should have Q(t) decreasing with time, not increasing. You should have:
$\displaystyle Q(t) = Q_0 +\frac {R_0} k (e^{-kt} -1)$
If you set t to infinity you get:
$\displaystyle Q(\infty) = Q_0 + \frac {R_0} k (e^{-\infty} -1) = Q_0 - \frac {R_0} k $
Set $\displaystyle Q(\infty) = 0$ to determine the minimum value of k in terms of $\displaystyle Q_0$ and $\displaystyle R_0$ that satisfies the condition.