  In the context of Born-Infeld gravity theories we report the existence of a
regular black hole interior representing a spherically symmetric vacuum
solution of the theory. It reduces to the Schwarzschild interior metric in the
weak field region. In particular, there is a new length scale which is related
to the Born-Infeld parameter. This endows the spacetime with an inner (i.e.
well inside the event horizon) asymptotic region which is unattainable for
observers. The central curvature singularity is replaced by an infinitely long
cosmic string with constant curvature invariants related to the Born-Infeld
constant. The presence of this limiting curvature spacetime renders the black
hole timelike and null geodesically complete, free from the classical
Schwarzschild singularity. The transition between the usual black hole interior
and this maximum curvature space is achieved without introducing any kind of
matter content nor topological changes.
