Context. We report on a successful, simultaneous observation and modelling of the millimeter (mm) to near-infrared (NIR) flare emission of the Sgr A* counterpart associated with the supermassive (4 × 10 6 M ) black hole at the Galactic centre (GC). We present a mm/sub-mm light curve of Sgr A* with one of the highest quality continuous time coverages. Aims. We study and model the physical processes giving rise to the variable emission of Sgr A*. Methods. Our non-relativistic modelling is based on simultaneous observations carried out in May 2007 and 2008, using the NACO adaptive optics (AO) instrument at the ESO's VLT and the mm telescope arrays CARMA in California, ATCA in Australia, and the 30 m IRAM telescope in Spain. We emphasize the importance of multi-wavelength simultaneous fitting as a tool for imposing adequate constraints on the flare modelling. We present a new method for obtaining concatenated light curves of the compact mm-source Sgr A* from single dish telescopes and interferometers in the presence of significant flux density contributions from an extended and only partially resolved source. Results. The observations detect flaring activity in both the mm domain and the NIR. Inspection and modelling of the light curves show that in the case of the flare event on 17 May 2007, the mm emission follows the NIR flare emission with a delay of 1.5 ± 0.5 h. On 15 May 2007, the NIR flare emission is also followed by elevated mm-emission. We explain the flare emission delay by an adiabatic expansion of source components. For two other NIR flares, we can only provide an upper limit to any accompanying mmemission of about 0.2 Jy. The derived physical quantities that describe the flare emission give a source component expansion speed of v exp ∼ 0.005c−0.017c, source sizes of about one Schwarzschild radius, flux densities of a few Janskys, and spectral indices of α = 0.6 to 1.3. These source components peak in the THz regime. Conclusions. These parameters suggest that either the adiabatically expanding source components have a bulk motion greater than v exp or the expanding material contributes to a corona or disk, confined to the immediate surroundings of Sgr A*. Applying the flux density values or limits in the mm-and X-ray domain to the observed flare events constrains the turnover frequency of the synchrotron components that are on average not lower than about 1 THz, such that the optically thick peak flux densities at or below these turnover frequencies do not exceed, on average, about ∼1 Jy.