Millimeter and submillimeter observations of 53 rotational transitions of 18 different molecules and their isotopic varieties are presented at five positions on a strip crossing the Orion Bar. These data are used to study the physical and chemical conditions with increasing depth in to the photon-dominated region (PDR). It is found that all molecules show about the same intensity distribution across the Bar. They peak approximately 30 arcsec outside the ionization front, and fall off to about 10-20% of their peak values on either side. An excitation analysis of C18O, H2CO, CS and HCO+ shows that the Orion Bar is best described by an inhomogeneous density distribution where approximately 10% of the molecular material is comprised in clumps of hydrogen number density n(H2) approximately equals 1(-0.7,+3.0) 10^6/cm3 and approximately 90% in a homogeneous interclump medium of n(H2) approximately equals 3(-2.2,+2.0) 10^4/cm3. The kinetic temperature is found to be 85 +/- 30 K. No significant variations in these physical parameters occur across the Bar. It is therefore concluded that the observed peak in the molecular rotational emission corresponds to a peak in molecular column densities. A geometric model for the Orion Bar is proposed in which with increasing distance from the ionizing stars, the PDR changes from a face-on to an edge-on geometry and back. The increased emission is then caused by the increased length of the line-of-sight through the PDR. This model is found to correspond well with other observations.