Context. Betelgeuse is the closest red supergiant (RSG); therefore, it is well suited for studying the complex processes in its atmosphere that lead to the chemical enrichment of the interstellar medium. Aims. We intend to investigate the shape and composition of the close molecular layer (also known as the MOLsphere) that surrounds the star. This analysis is part of a wider program that aims at understanding the dynamics of the circumstellar envelope of Betelgeuse. Methods. On January and February 2011, Betelgeuse was observed using the Astronomical Multi-BEam combineR (AMBER) instrument of the Very Large Telescope Interferometer (VLTI) in the H and K bands. Using the medium spectral resolution of the instrument (R $∼$ 1500), we were able to investigate the carbon monoxide band heads and the water-vapor bands. We used two different approaches to analyse our data: a model fit in both the continuum and absorption lines and then a fit with a Radiative HydroDynamics (RHD) simulation. Results. Using the continuum data, we derive a uniform disk diameter of $41.01 ± 0.41$~mas, a power law type limb-darkened disk diameter of $42.28 ± 0.43$~mas and a limb-darkening exponent of $0.155 ± 0.009$. Within the absorption lines, using a single layer model, we obtain parameters of the MOLsphere. Using a RHD simulation, we unveil the convection pattern in the visibilities. Conclusions. We derived a new value of the angular diameter of Betelgeuse in the K band continuum. Our observations in the absorption lines are well reproduced by a molecular layer at 1.2 stellar radii containing both CO and H$_2$O. The visibilities at higher spatial frequencies are matching a convection pattern in a RHD simulation. ; Comment: 13 pages, 11 figures, accepted for publication in Astronomy & Astrophysics; Language editing