The operator \(S^{\phi}f=\sum_{n\in\mathbb Z}\phi(n)T^nf\) associated to an invertible measure-preserving transformation \(T\) and a discrete singular kernel \(\phi\) generalizes the ergodic Hilbert transform (in which \(\phi(n)=1/n\)). The existence almost everywhere of the Hilbert transform was shown by M. Cotlar [Rev. Mat. Cuyana 1, 105–167 (1956; Zbl 0071.33402)], and this was later generalized in several directions. These generalizations involve additive processes; the author [Can. Math. Bull. 49, No. 2, 203–212 (2006; Zbl 1116.28012)] proved the almost everywhere existence for bounded symmetric ‘admissable’ processes under a condition on the purely subadditive part of the process. In this paper the additional condition is removed, and a general almost everywhere existence proof is given for discrete ergodic singular transforms for bounded symmetric admissable processes.