  We propose in this White Paper a concept for a space experiment using cold
atoms to search for ultra-light dark matter, and to detect gravitational waves
in the frequency range between the most sensitive ranges of LISA and the
terrestrial LIGO/Virgo/KAGRA/INDIGO experiments. This interdisciplinary
experiment, called Atomic Experiment for Dark Matter and Gravity Exploration
(AEDGE), will also complement other planned searches for dark matter, and
exploit synergies with other gravitational wave detectors. We give examples of
the extended range of sensitivity to ultra-light dark matter offered by AEDGE,
and how its gravitational-wave measurements could explore the assembly of
super-massive black holes, first-order phase transitions in the early universe
and cosmic strings. AEDGE will be based upon technologies now being developed
for terrestrial experiments using cold atoms, and will benefit from the space
experience obtained with, e.g., LISA and cold atom experiments in microgravity.
  This paper is based on a submission (v1) in response to the Call for White
Papers for the Voyage 2050 long-term plan in the ESA Science Programme. ESA
limited the number of White Paper authors to 30. However, in this version (v2)
we have welcomed as supporting authors participants in the Workshop on Atomic
Experiments for Dark Matter and Gravity Exploration held at CERN: ({\tt
https://indico.cern.ch/event/830432/}), as well as other interested scientists,
and have incorporated additional material.
