In this very important paper a linear time-invariant system, which is represented by a one-dimensional Euler-Bernoulli beam equation in a bounded domain, is considered. The beam is clamped at one end, and the boundary control force input is applied at the other end. For the system, a finite-dimensional dynamic boundary controller is proposed. The transfer function of the controller is a proper rational function of the complex variable \(s\) and may contain a single pole at \(s= 0\) and another one at \(s= j\omega_1\), \(\omega_1\neq 0\); provided that the residues corresponding to these poles are nonnegative, the rest of the transfer function is required to be a strictly positive real function. Main result: The author shows that the closed-loop system is asymptotically stable provided that \(s= j\omega_1\) is not the zero of an appropriate system transfer function and that it is exponentially stable in some cases. The case where the output of the controller is corrupted by a disturbance is analyzed. Moreover, it is shown that, if the frequency spectrum of the controller is known, then by choosing the controller appropriately we can obtain better disturbance rejection.