This is the first of two manuscripts dealing with the circulation, mixing, ventilation and organic matter mineralization of the South Atlantic Ocean (SAO). The present work quantifies the complex mixing of water masses in the SAO using a constrained, least-squares regression, Optimum MultiParameter (OMP) analysis. The OMP based on temperature, salinity, silicate and the conservative parameter NO, was applied on two World Ocean Circulation (WOCE) lines, A17 and A14, in the western and eastern SAO, respectively. The constrained OMP sensitivity to sources of error in the end-member characteristics, measured parameters, equation weights and oxygen to nitrogen mineralisation ratio (RN) was carefully assessed using perturbation tests. Perturbation of RN was the only test that changed significantly the mixing proportions although by less than 5%. The constrained OMP method allowed defining the realm and identifying the core-of-flow of each water mass to study its circulation, the evolution of its chemical composition and, eventually, to separate the contribution of physical and biogeochemical processes. Relevant specific outputs of this first manuscript are: (1) north of the South Equatorial Current, the silicate maximum is primarily composed of Antarctic Intermediate Water (AAIW) rather than Circumpolar Deep Water (CDW); (2) the two degree discontinuity (TDD) experiences a dramatic meridional change of water masses composition, being dominated by North Atlantic Deep Water (NADW) north of the Vema Channel and by CDW southwards; (3) the 50% proportion horizon of Weddell Sea Deep Water (WDSW), with a θ of -0.3°C at the entry of the SAO, defines the upper limit of the WSDW realm more properly than the classical isopycnals of 46.04 or 46.06 σ4, where the proportion of WSDW is around 75%.