We prove that given any meromorphic function / on a compact Riemann surface M' there exists another meromorphic function g on M' such that {df, g} is the Weierstrass pair defining a complete conformai minimal immersion of finite total curvature into Euclidean 3-space defined on M' punctured at a finite set of points. As corollaries we obtain i) any compact Riemann surface can be immersed in Euclidean 3-space as in the above with at most 4p + 1 punctures, where p is the genus of the Riemann

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... ce; ii) any hyperelliptic Riemann surface of genus p can be so immersed with at most 3p + 4 punctures.