  In order to gain a better understanding of the origin of decoherence in
superconducting flux qubits, we have measured the magnetic field dependence of
the characteristic energy relaxation time ($T_1$) and echo phase relaxation
time ($T_2^{\rm echo}$) near the optimal operating point of a flux qubit. We
have measured $T_2^{\rm echo}$ by means of the phase cycling method. At the
optimal point, we found the relation $T_2^{\rm echo}\approx 2T_1$. This means
that the echo decay time is {\it limited by the energy relaxation} ($T_1$
process). Moving away from the optimal point, we observe a {\it linear}
increase of the phase relaxation rate ($1/T_{2}^{\rm echo}$) with the applied
external magnetic flux. This behavior can be well explained by the influence of
magnetic flux noise with a $1/f$ spectrum on the qubit.
