  Let $F$ be a locally compact non-Archimedean field, and let $B/F$ be a
division algebra of dimension 4. The Jacquet-Langlands correspondence provides
a bijection between smooth irreducible representations of $B^\times$ of
dimension $>1$ and irreducible cuspidal representations of $GL_2(F)$. We
present a new construction of this bijection in which the preservation of
epsilon factors is automatic.
