  In materials without an inversion center of symmetry the spin degeneracy of
the conducting band is lifted by an antisymmetric spin orbit coupling (ASOC).
Under such circumstances, spin and parity cannot be separately used to classify
the Cooper pairing states. Consequently, the superconducting order parameter is
generally a mixture of spin singlet and triplet pairing states. In this paper
we investigate the structure of the order parameter and its response to
disorder for the most symmetric pairing state ($A_1$). Using the example of the
heavy Fermion superconductor CePt$_3$Si, we determine characteristic properties
of the superconducting instability as a function of (non-magnetic) impurity
concentrations. Moreover, we explore the possibility of the presence of
accidental line nodes in the quasiparticle gap. Such nodes would be essential
to explain recent low-temperature data of thermodynamic quantities such as the
NMR-$T_1^{-1}$, London penetration depth, and heat conductance.
