A solid—amorphous or crystalline—is defined by a finite shear modulus while a fluid lacks such. We thus experimentally investigate the elastic properties of a colloidal glass former near the glass transition: Spectroscopy of vibrational excitations yields the dispersion relations of longitudinal and transverse phonons in the glassy state. From the long-wavelength limit of the dispersion relation, we extract the bulk and the shear modulus. As expected, the latter disappear in a fluid and we measure a clearly resolved discontinuous behavior of the elastic moduli at the glass transition. This not only determines the transition temperature TG of the system but also directly addresses recent discussions about elasticity during vitrification. We show that low-frequency excitations in our system are plane waves such that continuum elasticity theory can be used to describe the macroscopic behavior.