  We present an exact solution for the probability density function
$P(\tau=t_{\min}-t_{\max}|T)$ of the time-difference between the minimum and
the maximum of a one-dimensional Brownian motion of duration $T$. We then
generalise our results to a Brownian bridge, i.e. a periodic Brownian motion of
period $T$. We demonstrate that these results can be directly applied to study
the position-difference between the minimal and the maximal height of a
fluctuating $(1+1)$-dimensional Kardar-Parisi-Zhang interface on a substrate of
size $L$, in its stationary state. We show that the Brownian motion result is
universal and, asymptotically, holds for any discrete-time random walk with a
finite jump variance. We also compute this distribution numerically for L\'evy
flights and find that it differs from the Brownian motion result.
