The Southern inTermediate Redshift ESO Supernova Search (STRESS) is aimed to measure the the Supernova rates at intermediate redshift. Unlike most of current Supernova searches, this survey was specifically designed to estimate the rate for both of Type Ia and Core Collapse Supernovae. We followed an approach in the Supernova rate measurement that allows us to explore the properties of the monitored galaxies and to link them to Supernova rates. Our analysis is based on a sample of 40000 galaxies and on the 26 spectroscopically confirmed Supernovae plus 64 reliable Supernova candidates detected monitoring our galaxy sample for about five years. Our results per unity B luminosity indicate that the Core Collapse Supernova rate increases at intermediate redshift with respect to local estimate of a factor 2 whereas the Type Ia Supernova rate remains almost constant. The observed trend of Supernova rates from ``red'' to ``blue'' galaxies at intermediate redshift is similar to that in the local Universe. We compare the evolution of the rates of both Supernova types with published measurements and find a good agreement. Assuming an initial mass function and the standard scenario for Core Collapse progenitors, we compare our estimate of rate evolution with the star formation history inferred through different star formation indicators obtaining interesting remarks. We compare also the observed evolution of the type Ia Supernova rate with redshift with prediction of different progenitor models, assuming a star formation history. Although the progenitor models are not constrained by measurements at intermediate redshifts, they predict a modest evolution of type Ia rate.