Let $(B_t ; t \ge 0)$, $\big(\mbox{resp. }((X_t, Y_t)
; t \ge 0)\big)$ be a one (resp. two) dimensional Brownian
motion started at 0. Let $T$ be a stopping time such that $(B_{t
\wedge T} ; t \ge 0)$ \big(resp. $(X_{t \wedge T} ; t \ge 0) ;
(Y_{t \wedge T} ; t \ge 0)\big)$ is uniformly integrable. The
main results obtained in the paper are:
\begin{itemize}
\item[1)] if $T$ and $B_T$ are independent and $T$ has all
exponential moments, then $T$ is constant.
\item[2)] If $X_T$ and $Y_T$ are independent and have all
exponential moments, then $X_T$ and $Y_T$ are Gaussian.
\end{itemize}
We also give a number of examples of stopping times $T$, with only
some exponential moments, such that $T$ and $B_T$ are independent,
and similarly for $X_T$ and $Y_T$. We also exhibit bounded
non-constant stopping times $T$ such that $X_T$ and $Y_T$ are
independent and Gaussian.