Abstract: We consider Brox's model: a one-dimensional diffusion in a Brownian environment. We show the weak convergence of the normalized local time process $(L(x+m_{\log t},t)/t,x\in I \subset \R)$, centered at the coordinate of the bottom of the deepest valley $m_{\log t}$ reached by the process before time $t$ to a functional of two independent 3-dimensional Bessel processes. We apply that result to get the limit law of the supremum of the normalized local time. These results are discussed and compared to the discrete time and space analogous model whose same questions have been solved recently by N. Ganter, Y. Peres and Z. Shi.