  A compact complex Hermitian manifold $(M, I, w)$ is called Vaisman if
$dw=w\wedge \theta$ and the 1-form $\theta$, called the Lee form, is parallel
with respect to the Levi-Civita connection. The volume form of $M$ is invariant
with respect to the action of the vector field $X$ dual to $\theta$ (called the
Lee field) and the vector field $I(X)$, called { the anti-Lee field}. The
cohomology class of $\theta$, called the Lee class, plays the same role as the
Kahler class in Kahler geometry. We prove that a Vaisman metric is uniquely
determined by its volume form and the Lee class, and, conversely, for each Lee
class $[\theta]$ and each Lee- and anti-Lee-invariant volume form $V$, there
exists a Vaisman structure with the volume form $V$ and the Lee class
$c[\theta]$. This is an analogue of the Calabi-Yau theorem claiming that the
Kahler form is uniquely determined by its volume and the cohomology class.
