Context. Betelgeuse is the closest red supergiant (RSG), therefore it is well suited to study 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 VLTI/AMBER interferometer in the H and K bands. Using the medium spectral resolution of the instrument ($R \sim 1500$), we were able to investigate the carbon monoxide band heads and the water-vapor bands. We used two different approaches to analyse our data: model fitting both in the continuum and absorption lines and then fit with a RHD simulation. Results. Using the continuum data we derive a uniform disk diameter of $41.01 \pm 0.41$ mas, a power law type limb-darkened disk diameter of $42.28 \pm 0.43$ mas and a limb-darkening exponent of $0.155 \pm 0.009$. 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.