We consider the problem of boundary feedback stabilization of a vibrating string that is fixed at one end and with control action at the other end. In contrast to previous studies that have required L-2-regularity for the initial position and H-1-regularity for the initial velocity, in this paper we allow for initial positions with L-1-regularity and initial velocities in W--1,W-1 on the space interval. It is well known that for a certain feedback parameter, for sufficiently regular initial states the classical energy of the closed-loop system with Neumann velocity feedback is controlled to zero after a finite time that is equal to the minimal time where exact controllability holds. In this paper, we present a Dirichlet boundary feedback that yields a well-defined closed-loop system in the (L-1, W--1,W-1) framework and also has this property. Moreover, for all positive feedback parameters our feedback law leads to exponential decay of a suitably defined L-1-energy. For more regular initial states with (L-2, H-1) regularity, the proposed feedback law leads to exponential decay of an energy that corresponds to this framework. If the initial states are even more regular with H-1-regularity of the initial position and L-2-regularity of the initial velocity, our feedback law also leads to exponential decay of the classical energy.