The natural night sky brightness is a relevant input for monitoring the light pollution evolution at observatory sites, by subtracting it from the overall sky brightness determined by direct measurements. It is also instrumental for assessing the expected darkness of the pristine night skies. The natural brightness of the night sky is determined by the sum of the spectral radiances coming from astrophysical sources, including zodiacal light, and the atmospheric airglow. The resulting radiance is modified by absorption and scattering before it reaches the observer. Therefore, the natural night sky brightness is a function of the location, time and atmospheric conditions. We present in this work GAMBONS (GAia Map of the Brightness Of the Natural Sky), a model to map the natural night brightness of the sky in cloudless and moonless nights. Unlike previous maps, GAMBONS is based on the extraatmospheric star radiance obtained from the Gaia catalogue. The Gaia-DR2 archive compiles astrometric and photometric information for more than 1.6 billion stars up to 𝐺 = 21magnitude. For the brightest stars, not included in Gaia-DR2, we have used the Hipparcos catalogue instead. After adding up to the star radiance the contributions of the diffuse galactic and extragalactic light, zodiacal light and airglow, and taking into account the effects of atmospheric attenuation and scattering, the radiance detected by ground-based observers can be estimated. This methodology can be applied to any photometric band, if appropriate transformations from the Gaia bands are available. In particular, we present the expected sky brightness for 𝑉 (Johnson), and visual photopic and scotopic passbands.