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Summary form only given. The strong coupling between an atom and the light field in an optical microresonator causes a single atom to alter the transmission of the cavity significantly, allowing its continuous observation. At the same time, the strong coupling leads to novel optical forces acting on the atom. It was predicted that it should be possible to trap an atom with single intracavity photons. Using an atomic fountain, we inject single ultracold neutral atoms in the standing-wave light field of a high-finesse optical cavity. When an atom is detected in the cavity, the laser light pumping the cavity is suddenly switched to a higher value. This increases the depth of the optical dipole potential and traps the atom in the cavity field.