Gold nano wires were mechanically fabricated in solution to study their quantized conductance behavior under the electrochemical potential control. A nano wire 1 nm in length showing a unit value of the quantized conductance (G₀ = 2e²/h) was fabricated in solution at room temperature by choosing appropriate electrochemical potential and electrolyte. As well as a unit value, several fractional values also appeared at the quantization. Intensity of the fractional conductance peak at 0.6 G₀ in the histogram increased as the electrochemical potential of the wire became negative. Conductance trace proves that the fractional peak in the histogram is due to reversible conductance fluctuation between a unit (1 G₀) and the fractional value (0.6 G₀). These experimental results strongly suggest the formation of a gold mono-atomic wire showing reversible phase transition between a dimerized state and an equal spacing state. Possibility regarding the mono-atomic wire as a predecessor for Pierls transition in 1 D metal system was discussed.