We constrain f(nu) identical with Omega(nu)/Omega(m), the fractional contribution of neutrinos to the total mass density in the Universe, by comparing the power spectrum of fluctuations derived from the 2 Degree Field Galaxy Redshift Survey with power spectra for models with four components: baryons, cold dark matter, massive neutrinos, and a cosmological constant. Adding constraints from independent cosmological probes we find f(nu)<0.13 (at 95% confidence) for a prior of 0.1<Omega(m)<0.5, and assuming the scalar spectral index n=1. This translates to an upper limit on the total neutrino mass m(nu,tot)<1.8 eV for "concordance" values of Omega(m) and the Hubble constant.