When can we achieve the desirable uniform exponential decay property of the vibration energy for two coupled vibrating strings with a stabilizer or damper installed at the coupling point? The author gives a complete answer for the problem. For the cases when the stabilizers are “symmetrically placed”, the system’s energy does not decay to zero with respect to time for some initial states. For all the other cases, if the proportion of the wave speeds on the two vibrating strings is a rational number, the semigroup satisfies the spectrum-determined growth assumption, so the uniform exponential decay property depends completely on the representative form of the ratio of the wave speeds. If the ratio is an irrational number, then the energy decays strongly for some cases, but the energy does not decay uniformly exponentially.