  We have studied the dephasing of a superconducting flux-qubit coupled to a
DC-SQUID based oscillator. By varying the bias conditions of both circuits we
were able to tune their effective coupling strength. This allowed us to measure
the effect of such a controllable and well-characterized environment on the
qubit coherence. We can quantitatively account for our data with a simple model
in which thermal fluctuations of the photon number in the oscillator are the
limiting factor. In particular, we observe a strong reduction of the dephasing
rate whenever the coupling is tuned to zero. At the optimal point we find a
large spin-echo decay time of $4 \mu s$.
