The paper is devoted to a local description of all conformal constant mean curvature immersions \(\Phi: D\to\mathbb{R}^3\) of \(\mathbb{C}\) or an open disc \(D\subset \mathbb{C}\). This description is based on the fact previously established by the authors that each such map can be produced in some canonical way from a meromorphic matrix-valued one form \[ \xi= \lambda^{-1} \left(\begin{matrix} 0 & f(z)\\ g(z) & 0\end{matrix} \right) \] (meromorphic potential). In the paper, necessary and sufficient conditions on \(f(z)\) to make \(\Phi\) a constant mean curvature immersion are given.