In this paper, the Mannheim mate curves of the proper biharmonic curves in Cartan--Vranceanu 3-dimensional spaces $(M,ds^2_{l,m})$, with $l^2\neq 4m$ and $m\neq0$ are studied. We give the definition of the Mannheim mate of a proper biharmonic curve and give the explicit parametric equations of that Mannheim mate curve in Cartan--Vranceanu 3-dimensional space. Moreover, we show that the distance between corresponding points of the Mannheim pairs is constant in Cartan--Vranceanu 3-dimensional spaces.