A fluid queue is a stochastic process which moves linearly with a rate that is determined by the state of a continuous-time Markov chain (CTMC). In this paper we construct an approximation to a fluid queue using a quasi birth-and-death process with rational arrival process components (QBD-RAP) and prove weak convergence via convergence of generators. The primary motivation for constructing the new approximation is to achieve a better approximation accuracy than existing methods while also ensuring that all approximations to probabilities have all probabilistic properties, such as non-negativity and bounded above by 1.