  We have investigated decoherence in Josephson-junction flux qubits. Based on
the measurements of decoherence at various bias conditions, we discriminate
contributions of different noise sources. In particular, we present a Gaussian
decay function of the echo signal as evidence of dephasing due to $1/f$ flux
noise whose spectral density is evaluated to be about $(10^{-6} \Phi_0)^2$/Hz
at 1 Hz. We also demonstrate that at an optimal bias condition where the noise
sources are well decoupled the coherence observed in the echo measurement is
mainly limited by energy relaxation of the qubit.
