  The heavy fermion compound YbRh$_2$Si$_2$ is studied by low-temperature
magnetization $M(T)$ and specific-heat $C(T)$ measurements at magnetic fields
close to the quantum critical point ($H_c=0.06$ T, $H\perp c$). Upon
approaching the instability, $dM/dT$ is more singular than $C(T)$, leading to a
divergence of the magnetic Gr\"uneisen ratio $\Gamma_{\rm mag}=-(dM/dT)/C$.
Within the Fermi liquid regime, $\Gamma_{\rm mag}=-G_r(H-H_c^{fit})$ with
$G_r=-0.30\pm 0.01$ and $H_c^{fit}=(0.065\pm 0.005)$ T which is consistent with
scaling behavior of the specific-heat coefficient in
YbRh$_2$(Si$_{0.95}$Ge$_{0.05}$)$_2$. The field-dependence of $dM/dT$ indicates
an inflection point of the entropy as a function of magnetic field upon passing
the line $T^\star(H)$ previously observed in Hall- and thermodynamic
measurements.
