We follow the evolution of fermion pairing in the dimensional crossover from three-dimensional to twodimensional as a strongly interacting Fermi gas of 6 Li atoms becomes confined to a stack of twodimensional layers formed by a one-dimensional optical lattice. Decreasing the dimensionality leads to the opening of a gap in radio-frequency spectra, even on the Bardeen-Cooper-Schrieffer side of a Feshbach resonance. The measured binding energy of fermion pairs closely follows the theoretical two-body binding energy and, in the two-dimensional limit, the zero-temperature mean-field BoseEinstein-condensation to Bardeen-Cooper-Schrieffer crossover theory. DOI: 10.1103/PhysRevLett.108.045302 PACS numbers: 67.85.Àd, 03.75.Lm, 05.30.Fk, 32.30.Bv Interacting fermions in coupled two-dimensional (2D) layers present unique physical phenomena and are central to the description of unconventional superconductivity in high-transition-temperature cuprates [1] and layered organic conductors [2]. Experiments on ultracold gases of fermionic atoms have allowed access to the crossover from Bose-Einstein condensation (BEC) of tightly bound fermion pairs to Bardeen-Cooper-Schrieffer (BCS) superfluidity of long-range Cooper pairs in three spatial dimensions [3,4] and, more recently, the confinement of interacting Fermi gases to two spatial dimensions [5][6][7][8][9]. A fermionic superfluid loaded into a periodic potential should form stacks of two-dimensional superfluids with tunable interlayer coupling [10][11][12][13], an ideal model for Josephsoncoupled quasi-2D superconductors [1,14]. For deep potentials in the regime of uncoupled 2D layers, increasing the temperature of the gas is expected to destroy superfluidity through the Berezinskii-Kosterlitz-Thouless mechanism [15][16][17], while more exotic multiplane vortex loop excitations are predicted for a three-dimensional (3D) anisotropic BCS superfluid near the critical point [18].In this Letter, we study fermion pairing across the crossover from 3D to 2D in a periodic potential of increasing depth. To form a bound state in 3D, the attraction between two particles in a vacuum must exceed a certain threshold. However, if the two particles interact in the presence of a Fermi sea, the Cooper mechanism allows pairing for arbitrarily weak interactions [19]. In 2D, even two particles in a vacuum can bind for arbitrarily weak interactions. Surprisingly, the mean-field theory of the BEC-BCS crossover in 2D predicts that the binding energy of fermion pairs in the many-body system is identical to the two-body binding energy E b [20]. Indeed, to break a pair and remove one pairing partner from the system costs an energy [21]À within mean-field theory, where is the chemical potential and Á is the pairing gap. In 2D, one finds [20] ¼ E F À E b =2 and Á 2 ¼ 2E F E b , where E F is the Fermi energy, and thus E b;MF ¼ E b ; i.e., the many-body and two-body binding energies are predicted to be identical throughout the BEC-BCS crossover.We realize a system that is tunable from 3D to 2D with a gas ...