We present phase coherence time measurements in quasi-one-dimensional mesoscopic wires made from high mobility two-dimensional electron gas. By implanting gallium ions into a GaAs/AlGaAs heterojunction we are able to vary the diffusion coefficient over 2 orders of magnitude. We show that in the diffusive limit, the decoherence time follows a power law as a function of diffusion coefficient as expected by theory. When the disorder is low enough so that the samples are semiballistic, we observe a new and unexpected regime in which the phase coherence time is independent of disorder. In addition, for all samples the temperature dependence of the phase coherence time follows a power law down to the lowest temperatures without any sign of saturation and this strongly suggests that the frequently observed low temperature saturation is not intrinsic.