Aims. In the course of a project to study eclipsing binary stars in vinicity of the Sun, we found that the cooler component of LL Aqr is a solar twin candidate. This is the first known star with properties of a solar twin existing in a non-interacting eclipsing binary, offering an excellent opportunity to fully characterise its physical properties with very high precision. Methods.We used extensive multi-band, archival photometry and the Super-WASP project and high-resolution spectroscopy obtained from the HARPS and CORALIE spectrographs. The spectra of both components were decomposed and a detailed LTE abundance analysis was performed. The light and radial velocity curves were simultanously analysed with the Wilson-Devinney code. The resulting highly precise stellar parameters were used for a detailed comparison with PARSEC, MESA, and GARSTEC stellar evolution models. Results. LL Aqr consists of two main-sequence stars (F9V + G3 V) with masses of M1 = 1.1949 +/- 0.0007 and M2 = 1.0337 +/- 0.0007 Msun, radii R1 = 1.321 +/- 0:006 and R2 = 1.002 +/- 0.005 Rsun, temperatures T1 = 6080 +/- 45 and T2 = 5703 +/- 50 K and solar chemical composition [M/H] = 0.02 +/- 0.05. The absolute dimensions, radiative and photometric properties, and atmospheric abundances of the secondary are all fully consistent with being a solar twin. Both stars are cooler by about 3.5sigma or less metal abundant by 5sigma than predicted by standard sets of stellar evolution models. When advanced modelling was performed, we found that full agreement with observations can only be obtained for values of the mixing length and envelope overshooting parameters that are hard to accept. The most reasonable and physically justified model fits found with MESA and GARSTEC codes still have discrepancies with observations but only at the level of 1sigma. The system is significantly younger that the Sun, with an age between 2.3 Gyr and 2.7 Gyr, which agrees well with the relatively high lithium abundance of the secondary, A(Li) = 1.65 +/- 0.10 dex.