We present results of ultrasonic measurements on a single crystal of the distorted diamond-chain compound azurite Cu$_3$(CO$_3$)$_2$(OH)$_2$. Pronounced elastic anomalies are observed in the temperature dependence of the longitudinal elastic mode $c_{22}$ which can be assigned to the relevant magnetic interactions in the system and their couplings to the lattice degrees of freedom. From a quantitative analysis of the magnetic contribution to $c_{22}$ the magneto-elastic coupling $G$ = $\partial J_2$/$\partial ε_b$ can be determined, where $J_2$ is the intra-dimer coupling constant and $ε_b$ the strain along the intra-chain $b$ axis. We find an exceptionally large coupling constant of $|G| \sim ($3650 $\pm$ 150) K highlighting an extraordinarily strong sensitivity of $J_2$ against changes of the $b$-axis lattice parameter. These results are complemented by measurements of the hydrostatic pressure dependence of $J_2$ by means of thermal expansion and magnetic susceptibility measurements performed both at ambient and finite hydrostatic pressure. We propose that a structural peculiarity of this compound, in which Cu$_2$O$_6$ dimer units are incorporated in an unusually stretched manner, is responsible for the anomalously large magneto-elastic coupling.

9 pages, 6 figures