We present a new strategy to couple, in a non-split fashion, stiff integration schemes with explicit, extended-stability predictor-corrector methods. The approach is illustrated through the construction of a mixed scheme incorporating a stabilised second-order, Runge-Kutta-Chebyshev method and the CVODE stiff solver. The scheme is first applied to an idealised stiff reaction-diffusion problem that admits an analytical solution. Analysis of the computations reveals that as expected the scheme exhibits a second-order in time convergence, and that, compared to an operator-split construction, time integration errors are substantially reduced. The non-split scheme is then applied to model the transient evolution of a freely-propagating, 1D methane-air flame. A low-mach-number, detailed kinetics, combustion model, discretised in space using fourth-order differences, is used for this purpose. To assess the performance of the scheme, self-convergence tests are conducted, and the results are contrasted with computations performed using a Strang-split construction. Whereas both the split and non-split approaches exhibit second-order in time behaviour, it is seen that for the same value of the time step, the non-split approach exhibits significantly smaller time integration errors. On the other hand, the results also indicate that the application of the present non-split construction becomes attractive when large integration steps are used, due to numerical overhead.