  We report on the Doppler-free saturation spectroscopy of the nitrous oxide
(N$_2$O) overtone transition at 1.28~$\mu$m. This measurement is performed by
the noise-immune cavity-enhanced optical heterodyne molecular spectroscopy
(NICE-OHMS) technique based on the quantum-dot (QD) laser. A high intra-cavity
power, up to 10~W, reaches the saturation limit of the overtone line using an
optical cavity with a high finesse of 113,500. At a pressure of several mTorr,
the saturation dip is observed with a full width at half-maximum of about 2~MHz
and a signal-to-noise ratio of 71. To the best of our knowledge, this is the
first saturation spectroscopy of molecular overtone transitions in 1.3~$\mu$m
region. The QD laser is then locked to this dispersion signal with a stability
of 15 kHz at 1 sec integration time. We demonstrate the potential of the N$_2$O
as markers because of its particularly rich spectrum at the vicinity of
1.28-1.30 $\mu$m where lies several important forbidden transitions of atomic
parity violation measurements and the 1.3 $\mu$m O-band of optical
communication.
