The Hilbert hotel is an old mathematical paradox about sets of infinite numbers. This paradox deals with the accommodation of a new guest in a hotel with an infinite number of occupied rooms. Over the past decade, there have been many attempts to implement these ideas in photonic systems. In addition to the fundamental interest that this paradox has attracted, this research is motivated by the implications that the Hilbert hotel has for quantum communication and sensing. In this work, we experimentally demonstrate the fractional vortex Hilbert's hotel scheme proposed by G. Gbur [Optica 3, 222-225 (2016)]. More specifically, we performed an interference experiment using the fractional orbital angular momentum of light to verify the Hilbert's infinite hotel paradox. In our implementation, the reallocation of a guest in new rooms is mapped to interference fringes that are controlled through the topological charge of an optical beam.