We give an explicit construction of the factorizing twists for the Yangian Y(₂) in evaluation representations (not necessarily finite dimensional). The result is a universal expression for the factorizing twist that holds in all of these representations. The method is general enough to recover the universal R-matrix of Y(₂) up to its character in the form specialized to generic evaluation representations. The method presented here is particularly amenable to generalizations because it involves only elementary operations applied to representations of the Yangian.