We extend recent work concerning isospectral deformations for one-dimensional Schrödinger operators to the case of Jacobi operators. We provide a complete spectral characterization of a new method that constructs isospectral deformations of a given Jacobi operator (H u)(n) = a(n) u(n+1) + a(n-1) u(n-1) - b(n) u(n). Our technique is connected to Dirichlet data, that is, the spectrum of the operator H∞n0 on l2 (-∞,n0) ⊕ l2 (n0,∞) with a Dirichlet boundary condition at n0. The transformation moves a single eigenvalue of H∞n0 and perhaps flips which side of n0 the eigenvalue lives. On the remainder of the spectrum the transformation is realized by a unitary operator.