We present direct measurements of a new geometric phase acquired by optical beams carrying orbital angular momentum. This phase arises when the transverse mode of a beam is transformed following a closed path in the space of modes. The measurements were done via the interference of two copropagating optical beams that pass through the same interferometer parts but acquire different geometric phases. The method is insensitive to dynamical phases. The magnitude and sign of the measured phases are in excellent agreement with theoretical predictions. © 2003 The American Physical Society.