We evaluate one-loop quantum gravity corrections to the conformally coupled (CC) scalar self-mass-squared on a locally de Sitter background. In this paper we consider only the conformal-conformal interaction part of the self-mass-squared. This complements the minimal-minimal part worked out in the previous paper [E. O. Kahya and R. P. Woodard, Phys. Rev. D 76, 124005 (2007)], and we add the minimal-conformal part in a follow-up paper to complete the full self-mass-squared at one-loop order. The computation is performed using dimensional regularization, and the results are fully renormalized by absorbing divergences with counterterms. The finite results can give rise to quantum corrections to the CC scalar mode functions and therefore to their power spectra.