  A key aspect of unconventional pairing by the antiferromagnetic
spin-fluctuation mechanism is that the superconducting energy gap must have
opposite sign on different parts of the Fermi surface. Recent observations of
non-nodal gap structure in the heavy-fermion superconductor CeCu$_2$Si$_2$ were
then very surprising, given that this material has long been considered a
prototypical example of a superconductor where the Cooper pairing is
magnetically mediated. Here we present a study of the effect of controlled
point defects, introduced by electron irradiation, on the temperature-dependent
magnetic penetration depth $\lambda(T)$ in CeCu$_2$Si$_2$. We find that the
fully-gapped state is robust against disorder, demonstrating that low-energy
bound states, expected for sign-changing gap structures, are not induced by
nonmagnetic impurities. This provides bulk evidence for $s_{++}$-wave
superconductivity without sign reversal.
