  Investigating the role of causal order in quantum mechanics has recently
revealed that the causal distribution of events may not be a-priori
well-defined in quantum theory. While this has triggered a growing interest on
the theoretical side, creating processes without a causal order is an
experimental task. Here we report the first decisive demonstration of a process
with an indefinite causal order. To do this, we quantify how incompatible our
set-up is with a definite causal order by measuring a 'causal witness'. This
mathematical object incorporates a series of measurements which are designed to
yield a certain outcome only if the process under examination is not consistent
with any well-defined causal order. In our experiment we perform a measurement
in a superposition of causal orders - without destroying the coherence - to
acquire information both inside and outside of a 'causally non-ordered
process'. Using this information, we experimentally determine a causal witness,
demonstrating by almost seven standard deviations that the experimentally
implemented process does not have a definite causal order.
