We investigate the state observer design problem for thermoacoustic instabilities in a Rijke tube using an infinite dimensional perspective. The observer, whose design is based on the backstepping methodology with Volterra and full integral terms, consists of a copy of the linearized plant model plus output injection terms and relies only on a single boundary acoustic pressure sensor. The exponential convergence, in the L-2 sense, of the observed error dynamics is proved, and the analytic expression of the observer gains are derived explicitly. The results are tested in an experimental Rijke tube prototype in order to illustrate the effectiveness of the method.