Discontinuous changes of the lattice parameters at the Mott metal-insulator transition are detected by high-resolution dilatometry on deuterated crystals of the layered organic conductor $kappa$-(BEDT-TTF)$_{2}$Cu[N(CN)$_{2}$]Br. The uniaxial expansivities uncover a striking and unexpected anisotropy, notably a zero-effect along the in-plane c-axis along which the electronic interactions are relatively strong. A huge thermal expansion anomaly is observed near the end-point of the first-order transition line enabling to explore the critical behavior with very high sensitivity. The analysis yields critical fluctuations with an exponent $tilde{alpha} simeq$ 0.8 $pm$ 0.15 at odds with the novel criticality recently proposed for these materials [Kagawa textit{et al.}, Nature textbf{436}, 534 (2005)]. Our data suggest an intricate role of the lattice degrees of freedom in the Mott transition for the present materials.