  Mechanical resonators based on a single carbon nanotube are exceptional
sensors of mass and force. The force sensitivity in these ultra-light
resonators is often limited by the noise in the detection of the vibrations.
Here, we report on an ultra-sensitive scheme based on a RLC resonator and a
low-temperature amplifier to detect nanotube vibrations. We also show a new
fabrication process of electromechanical nanotube resonators to reduce the
separation between the suspended nanotube and the gate electrode down to $\sim
150$~nm. These advances in detection and fabrication allow us to reach
$0.5~\mathrm{pm}/\sqrt{\mathrm{Hz}}$ displacement sensitivity. Thermal
vibrations cooled cryogenically at 300~mK are detected with a signal-to-noise
ratio as high as 17~dB. We demonstrate $4.3~\mathrm{zN}/\sqrt{\mathrm{Hz}}$
force sensitivity, which is the best force sensitivity achieved thus far with a
mechanical resonator. Our work is an important step towards imaging individual
nuclear spins and studying the coupling between mechanical vibrations and
electrons in different quantum electron transport regimes.
