Rydberg atoms have an electron in a state with a very high principal quantum number, and as a result can exhibit unusually long-range interactions. One example is the bonding of two such atoms by multipole forces to form Rydberg-Rydberg molecules with very large internuclear distances. Notably, bonding interactions can also arise from the low-energy scattering of a Rydberg electron with negative scattering length from a ground-state atom. In this case, the scattering-induced attractive interaction binds the ground-state atom to the Rydberg atom at a well-localized position within the Rydberg electron wavefunction and thereby yields giant molecules that can have internuclear separations of several thousand Bohr radii. Here we report the spectroscopic characterization of such exotic molecular states formed by rubidium Rydberg atoms that are in the spherically symmetric s state and have principal quantum numbers, n, between 34 and 40. We find that the spectra of the vibrational ground state and of the first excited state of the Rydberg molecule, the rubidium dimer Rb(5s)-Rb(ns), agree well with simple model predictions. The data allow us to extract the s-wave scattering length for scattering between the Rydberg electron and the ground-state atom, Rb(5s), in the low-energy regime (kinetic energy, <100meV), and to determine the lifetimes and the polarizabilities of the Rydberg molecules. Given our successful characterization of s-wave bound Rydberg states, we anticipate that p-wave bound states, trimer states and bound states involving a Rydberg electron with large angular momentum-so-called trilobite molecules-will also be realized and directly probed in the near future.