Precious little of today's mathematics would be possible without the real numbers. Mathematicians all learn about one or both of Dedekind cuts or Cantor's use of Cauchy sequences as nineteenth century constructions of the reals. These constructions, as part of the "arithmetization of analysis", allowed mathematics to move forward, by being assured of the existence and properties of the reals. They put on a firm foundation efforts to forge ahead with a theory of functions, in which rigorous analysis could replace naive geometry and intuition. It turns out, though, that the constructions of Richard Dedekind and Georg Cantor were neither the only, nor the first, such edifices representing the real numbers. It was Karl Weierstrass who provided the very first construction, particularly attuned to his interest in developing function theory via infinite series. Weierstrass never published his construction. I will present his construction and its properties in the context of a somewhat modern perspective, and discuss its advantages and disadvantages. The presentation will be accessible to anyone with an undergraduate analysis background.