  We have observed the transversal vibration mode of suspended carbon nanotubes
at millikelvin temperatures by measuring the single-electron tunneling current.
The suspended nanotubes are actuated contact-free by the radio frequency
electric field of a nearby antenna; the mechanical resonance is detected in the
time-averaged current through the nanotube. Sharp, gate-tuneable resonances due
to the bending mode of the nanotube are observed, combining resonance
frequencies of up to \nu_0 = 350 MHz with quality factors above Q = 10^5, much
higher than previously reported results on suspended carbon nanotube
resonators. The measured magnitude and temperature dependence of the Q-factor
shows a remarkable agreement with the intrinsic damping predicted for a
suspended carbon nanotube. By adjusting the RF power on the antenna, we find
that the nanotube resonator can easily be driven into the non-linear regime.
