This paper studies the dynamic behavior of a one-dimensional thermoelastic system with memory type in terms of its eigenfrequencies. The asymptotic expansions for eigenvalues and eigenfunctions are derived. It is shown that there is a sequence of generalized eigenfunctions, which forms a Riesz basis for the Hilbert state space. From this, we deduce the spectrum-determined growth condition, and as a consequence, the exponential stability of the system is concluded.