Runge, C., Über die Darstellung willkürlicher Functionen, Acta Math. 7 (1885/86), 387-392. This paper contains a proof of the fact that any continuous function on a finite interval can be uniformly approximated by rational functions. Phragmén (see footnote in Mittag-Leffler's 1900 paper) pointed out that Runge's previous article (Acta Math. 6) contains a method of replacing the rational functions by polynomials. These papers do not explicitly contain Weierstrass' Theorem.