  The theory of general relativity describes macroscopic phenomena driven by
the influence of gravity while quantum mechanics brilliantly accounts for
microscopic effects. Despite their tremendous individual success, a complete
unification of fundamental interactions is missing and remains one of the most
challenging and important quests in modern theoretical physics. The STE-QUEST
satellite mission, proposed as a medium-size mission within the Cosmic Vision
program of the European Space Agency (ESA), aims for testing general relativity
with high precision in two experiments by performing a measurement of the
gravitational redshift of the Sun and the Moon by comparing terrestrial clocks,
and by performing a test of the Universality of Free Fall of matter waves in
the gravitational field of Earth comparing the trajectory of two Bose-Einstein
condensates of Rb85 and Rb87. The two ultracold atom clouds are monitored very
precisely thanks to techniques of atom interferometry. This allows to reach
down to an uncertainty in the E\"otv\"os parameter of at least 2x10E-15. In
this paper, we report about the results of the phase A mission study of the
atom interferometer instrument covering the description of the main payload
elements, the atomic source concept, and the systematic error sources.
