We address the phenomenon of sedimentation of opinions in networks and investigate how stubborn agents who never change their minds ("zealots") can influence the opinion of a social group. This is done via the voter model in which users are divided in two camps and repeatedly update their opinions based on others they connect with. Assuming zealots are present on both sides, the distribution of opinions reaches an equilibrium. We give novel formulas based on Markov Chain analysis to compute the distribution of opinions over time and speed of convergence to stationary equilibrium. As an example of application we discuss a strategy to mitigate the polarisation of opinions in the presence of a backfire effect. Theoretical results are supported by numerical experiments on synthetic data.