We investigate Lyapunov exponents of Brownian motion in a nonnegative Poissonian potential \(V\). The Lyapunov exponent depends on the potential \(V\) and our interest lies in the decay rate of the Lyapunov exponent if the potential \(V\) tends to zero. In our model the random potential \(V\) is generated by locating at each point of a Poisson point process with intensity \(\nu\) a bounded compactly supported nonnegative function \(W\). We show that for sequences of potentials \(V_n\) for which \(\nu_n \|W_n\|_1 \sim D/n\) for some constant \(D > 0\) (\(n \to \infty\)), the decay rates to zero of the quenched and annealed Lyapunov exponents coincide and equal \(c n^{-1/2}\) where the constant \(c\) is computed explicitly. Further we are able to estimate the quenched Lyapunov exponent norm from above by the corresponding norm for the averaged potential.