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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 ${\pi }^{\prime }$ of $B^{\times }$ of dimension $\>1$ and irreducible cuspidal representations of ${GL}_2\left(F\right)$. We present a new construction of this bijection in which the preservation of epsilon factors is automatic. This is done by constructing a family of pairs $(ℒ,\rho )$, where $ℒ\subset M_2\left(F\right)\times B$ is an order and $\rho$ is a finite-dimensional representation of a certain...