  Bose-Einstein condensation is unique among phase transitions between
different states of matter in the sense that it occurs even in the absence of
interactions between particles. In Einstein's textbook picture of an ideal gas,
purely statistical arguments set an upper bound on the number of particles
occupying the excited states of the system, and condensation is driven by this
saturation of the quantum vapour. Dilute ultracold atomic gases are celebrated
as a realisation of Bose-Einstein condensation in close to its purely
statistical form. Here we scrutinise this point of view using an ultracold gas
of potassium (39K) atoms, in which the strength of interactions can be tuned
via a Feshbach scattering resonance. We first show that under typical
experi-mental conditions a partially condensed atomic gas strongly deviates
from the textbook concept of a saturated vapour. We then use measurements at a
range of interaction strengths and temperatures to extrapolate to the
non-interacting limit, and prove that in this limit the behaviour of a Bose gas
is consistent with the saturation picture. Finally, we provide evidence for the
universality of our observations through additional measurements with a
different atomic species, 87Rb. Our results suggest a new way of characterising
condensation phenomena in different physical systems.

Quantum Gases;Statistical Mechanics;Atomic Physics;Quantum Physics