The observation of a new, polar class of homonuclear diatomic molecules, called trilobite molecules, is presented. The molecules have permanent electric dipole moments of &sim20-100 Debye. The observations are in agreement with calculations carried out by our collaborators at the Institute of Theoretical Atomic, Molecular, and Optical Physics (ITAMP), at Harvard University. The unique mechanism that binds the molecules will be described. The molecules are not observable inside of a Magneto-Optical Trap (MOT) due to the low density of trapped atoms, &sim1 &times1010 cm-3. This thesis also describes the improvements to the apparatus in an effort to increase the density of trapped atoms. The improvements are two-fold. First, a system to slow an atomic beam using the Zeeman effect is described. The slowed atomic beam is used to load the MOT instead of from a background vapor, enhancing both the loading rate and the number of trapped atoms. A Monte Carlo simulation of the slowing process is presented. Second, an optical dipole trapping system has been developed and successfully implemented which captures atoms at a temperature of &sim40 &muK and at a density of &sim 2 &times1013 cm-3. The increase in density augments two-body event rates by a factor of &sim4 &times106, and allows experiments to probe smaller interaction distances. This is demonstrated by the photoassociation of these exotic trilobite molecules.