  We consider the class of bounded symmetric Jacobi matrices $J$ with positive
off-diagonal elements and complex diagonal elements. With each matrix $J$ from
this class, we associate the spectral data, which consists of a pair
$(\nu,\psi)$. Here $\nu$ is the spectral measure of $|J|=\sqrt{J^*J}$ and
$\psi$ is a $\textit{phase function}$ on the real line satisfying $|\psi|\leq1$
almost everywhere with respect to the measure $\nu$. Our main result is that
the map from $J$ to the pair $(\nu,\psi)$ is a bijection between our class of
Jacobi matrices and the set of all spectral data.
