The initial-value problem (IVP) for small perturbations is formulated for the linear stability of an idealized, planar Chapman-Jouguet (CJ) detonation. The analysis of the IVP requires consideration of the asymptotic solutions of both direct and adjoint stability equations. Previously, Erpenbeck (Phys. Fluids, Vol. 5, No. 1962, pp. 604-614) and Tumin (Phys. Fluids, Vol. 19, No. 10, 2007) had considered the IVP for an overdriven detonation. The CJ case requires a more delicate analysis since a sonic point exists downstream of the shock which represents a singularity in the stability equations. The asymptotic solutions near this singularity of the direct problem were provided by Sharpe (PRSL A, Vol. 453, 1997, pp. 2603-2625) and Short et al. (JFM, Vol. 595, 2008, pp. 45-82) within the context of the normal mode approach for an idealized gaseous and condensed-phase model of detonation, respectively. In the present work, the asymptotic analysis of the adjoint problem is completed and the full solution to the IVP for the CJ case is obtained. The result shows that the obtained solution has the same structure as in the overdriven detonation.