Single-cell recordings of transcriptional and post-transcriptional processes reveal the inherent stochasticity of cellular events. However, to a large extent, the observed variability in isogenic cell populations is due to extrinsic factors, such as difference in expression capacity, cell volume and cell cycle stageto name a few. Thus, such experimental data represents a convolution of effects from stochastic kinetics and extrinsic noise sources. Recent parameter inference schemes for single-cell data just account for variability because of molecular noise. Here, we present a Bayesian inference scheme that deconvolutes the two sources of variability and enables us to obtain optimal estimates of stochastic rate constants of low copy-number events and extract statistical information about cell-to-cell variability. In contrast to previous attempts, we model extrinsic noise by a variability in the abundance of mass-conserved species, rather than a variability in kinetic parameters. We apply the scheme to a simple model of the osmostress-induced transcriptional activation in budding yeast.