Aims. We study the radio emission mechanism of the FR-I AGN NGC 5128 (Centaurus A). Methods. We determine the centimeter and millimeter continuum spectrum of the whole Centaurus A radio source and measure at frequencies between 86 GHz (3.5 mm) and 345 GHz (0.85 mm) the continuum emission from the active radio galaxy nucleus at various times between 1989 and 2005. Results. The integral radio source spectrum becomes steeper at frequencies above 5 GHz, where the spectral index changes from $\alpha_{\rm low}$ = -0.70 to $\alpha_{\rm high}$ = -0.82. The SW outer lobe has a steeper spectrum than the NE middle and outer lobes (α = -1.0 vs. -0.6). Millimeter emission from the core of Centaurus A is variable, a variability that correlates appreciably better with the 20-200 keV X-ray variability than with 2-10 keV variability. Conclusions. In its quiescent state, the core has a spectral index α = -0.3, which steepens when the core brightens. The variability appears to be mostly associated with the inner nuclear jet components that have been detected in VLBI measurements. The densest nuclear components are optically thick below 45-80 GHz.