The isotope shift between singly-charged ¹⁵¹Eu and ¹⁵³Eu in the 4f⁷(⁸S^o)6s ⁹S^o₄ - 4f⁷(⁸S^o)⁶p_1/2 J = 4 transition at 4129 Å has been measured using fast ion beam-laser technique. This Eu line has attracted interest in connection with efforts of obtaining a cosmochronometer based on observed Th/Eu abundance ratios. Knowledge of the isotope shift is of importance in order to check that contaminations from line blends do not contribute to the line intensity of Eu II. The measured value of the isotope shift -0.1527 (2) cm^-1 is consistent with the old spectroscopic value of Krebs and Winkler (-0.1503 (25)cm^-1) using a Fabry-Perot interferometer, while the accuracy is improved substantially.