The Aria project consists of a plant, hosting a 350 m cryogenic isotopic distillation column, the tallest ever built, which is currently in the installation phase in a mine shaft at Carbosulcis S.p.A., Nuraxi-Figus (SU), Italy. Aria is one of the pillars of the argon dark-matter search experimental program, lead by the Global Argon Dark Matter Collaboration. Aria was designed to reduce the isotopic abundance of $^{39}$Ar, a $\beta$-emitter of cosmogenic origin, whose activity poses background and pile-up concerns in the detectors, in the argon used for the dark-matter searches, the so-called Underground Argon (UAr). In this paper, we discuss the requirements, design, construction, tests, and projected performance of the plant for the isotopic cryogenic distillation of argon. We also present the successful results of isotopic cryogenic distillation of nitrogen with a prototype plant, operating the column at total reflux.