This is the first paper of the GEMS (Gas phase Elemental abundances in Molecular CloudS) project, dedicated to the study of the prototypical dark cloud TMC1. GEMS is an IRAM 30m Large Programme aimed to determine the S, C, N, O depletions and X(e-) as a function of visual extinction in a selected set of prototypical star forming regions. Extensive millimeter observations have been carried out with the IRAM 30m telescope (3mm and 2mm) and the 40m Yebes telescope (bands K and Q) to determine the abundances of CO, HCO$^+$, HCN, CS, SO, HCS$^+$ and N$_{2}$H$^+$ in three cuts across the TMC1 filament. These cuts intersect the dense filament at the well-known positions TMC1-CP, TMC1-NH3 and TMC1-C and cover a visual extinction range of $\sim$3 mag to $>$20 mag. Two phases with differentiated physical conditions and chemistry can be distinguished in TMC1: i) the translucent phase that is characterized by molecular hydrogen densities of 1-5x10$^{-3}$ cm$^{-3}$ and is located at visual extinctions of 3--7 mag; ii) the dense phase with molecular hydrogen densities of a few 10$^4$ cm$^{-3}$, located at A$_v$~$>$~10 mag. The transition between these two phases (from 7 to 10 mag) occurs in a spatial scale of $<$0.04 pc and it is not well sampled by our observations. We determine the S, C, N, O depletions through the comparison of our molecular abundance estimates with the Meudon PDR code. Our results show that carbon, oxygen and sulfur are significantly depleted in the CII/C/CO transition zone (Av~$\sim$~3-4~mag) with C/H~$\sim$~8x10$^{-5}$, C/O~$\sim$~1, and S/H~$\sim$~8x10$^{-7}$. The gas phase abundances of C and O decrease by an additional factor of $\sim$2 during the translucent phase until the beginning of the dense phase at $\sim$10 mag. During the translucent phase S/H remains constant within the uncertainty of our measurements (a factor of 2).< Réduire