In a previous paper (AMI Consortium 2011, MNRAS, 415, 2699: Paper I), the observational, mapping and source-extraction techniques used for the Tenth Cambridge (10C) Survey of Radio Sources were described. Here, the first results from the survey, carried out using the Arcminute Microkelvin Imager (AMI) Large Array (LA) at an observing frequency of 15.7 GHz, are presented. The survey fields cover an area of ~ 27 deg² to a flux-density completeness of 1 mJy. Results for some deeper areas, covering ~ 12 deg², which are wholly contained within the total areas and complete to 0.5 mJy, are also presented. The completeness for both areas is estimated to be at least 93 per cent. The 10C survey is the deepest radio survey of any significant extent (>~ 0.2 deg²) above 1.4 GHz. The 10C source catalogue contains 1897 entries detected above a flux density threshold of > 4.62 sigma, and is available here and at the authors' web site <a href="http://www.mrao.cam.ac.uk/surveys/10C">http://www.mrao.cam.ac.uk/surveys/10C</a>;. The source catalog has been combined with that of the Ninth Cambridge Survey to calculate the 15.7-GHz source counts. A broken power law is found to provide a good parametrization of the differential count between 0.5 mJy and 1 Jy. The measured source count has been compared with that predicted by de Zotti et al. (2005, A&A, 431, 893, and the model is found to display good agreement with the data at the highest flux densities. However, over the entire flux-density range of the measured count (0.5 mJy to 1 Jy), the model is found to underpredict the integrated count by ~ 30 per cent. Entries from the source catalog have been matched with those contained in the catalogues of the NRAO VLA Sky Survey and the Faint Images of the Radio Sky at Twenty-cm survey (both of which have observing frequencies of 1.4 GHz). This matching provides evidence for a shift in the typical 1.4-GHz spectral index to 15.7-GHz spectral index of the 15.7-GHz-selected source population with decreasing flux density towards sub-mJy levels - the spectra tend to become less steep. Automated methods for detecting extended sources, developed in Paper I, have been applied to the data; ~ 5 per cent of the sources are found to be extended relative to the LA-synthesized beam of ~ 30 arcsec. Investigations using higher resolution data showed that most of the genuinely extended sources at 15.7 GHz are classical doubles, although some nearby galaxies and twin-jet sources were also identified. This table was created by the HEASARC in August 2011 based on an electronic version of Table 1 of the reference paper which was obtained from the 10C Survey web site <a href="http://www.mrao.cam.ac.uk/surveys/10C/">http://www.mrao.cam.ac.uk/surveys/10C/</a>;. This is a service provided by NASA HEASARC .