With control input constraints, an adaptive control algorithm is proposed for a plant which is subject to unmodeled dynamics. By introducing an additional feedback signal, it is shown that the uniform boundedness of all signals in the adaptive loop is preserved. The true nominal plant is assumed to be stable but unnecessarily minimum phase. Moreover, various properties of this algorithm are analyzed. It is also shown that the performance which can be achieved under no control input constraint in the nonadaptive case (i.e. the case that the true nominal plant is known as a priori) is asymptotically well maintained in the adaptive system. The analysis is supported by simulation results.