  This paper presents the results of specific-heat and magnetization
measurements, in particular their field-orientation dependence, on the first
discovered heavy-fermion superconductor CeCu$_2$Si$_2$ ($T_{\rm c} \sim 0.6$
K). We discuss the superconducting gap structure and the origin of the
anomalous pair-breaking phenomena, leading e.g., to the suppression of the
upper critical field $H_{\rm c2}$, found in the high-field region. The data
show that the anomalous pair breaking becomes prominent below about 0.15 K in
any field direction, but occurs closer to $H_{\rm c2}$ for $H \parallel c$. The
presence of this anomaly is confirmed by the fact that the specific-heat and
magnetization data satisfy standard thermodynamic relations. Concerning the gap
structure, field-angle dependences of the low-temperature specific heat within
the $ab$ and $ac$ planes do not show any evidence for gap nodes. From
microscopic calculations in the framework of a two-band full-gap model, the
power-law-like temperature dependences of $C$ and $1/T_1$, reminiscent of nodal
superconductivity, have been reproduced reasonably. These facts further support
multiband full-gap superconductivity in CeCu$_2$Si$_2$.
