We consider feedback boundary control of hyperbolic systems with stiff source terms. By combining weighted Lyapunov functions, the structure is used to derive stabilisation results. In our analysis, we give stabilising feedback laws that allow a robust uniform exponential stabilisation for a whole range of values of an uncertain parameter with a decay rate that is independent of the parameter. In particular, we are interested in the limit case when the relaxation parameter approaches zero. The result is illustrated with the numerical analysis on the decay rate of the Lyapunov function in terms of the stiff parameter and an application to boundary stabilisation of gas dynamics in pipes.