  Pitman's theorem states that if {Bt, t $\ge$ 0} is a one-dimensional Brownian
motion, then {Bt -- 2 inf s$\le$t Bs, t $\ge$ 0} is a three dimensional Bessel
process, i.e. a Brownian motion conditioned in Doob sense to remain forever
positive. In this paper one gives a similar representation for the Brownian
motion in an interval. Due to the double barrier, it is much more involved and
only asymptotic. This uses the fact that the interval is an alcove of the
Affine Lie algebra A 1 1 .
