A suspended, doubly clamped single wall carbon nanotube is characterized as driven nano-electromechanical resonator at cryogenic temperatures. Electronically, the carbon nanotube displays small bandgap behavior with Coulomb blockade oscillations in electron conduction and transparent contacts in hole conduction. We observe the driven mechanical resonance in d.c.-transport, including multiple higher harmonic responses. The data show a distinct negative frequency tuning at finite applied gate voltage, enabling us to electrostatically decrease the resonance frequency to 75% of its maximum value. This is consistently explained via electrostatic softening of the mechanical mode. (C) 2013 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim