This paper describes a study of the spin-up of a free-surface fluid in a rectangular container in which an internal cylindrical obstacle is mounted. Laboratory experiments have been carried out for a variety of obstacle positions. It was found that the flow evolution during the adjustment process leading to the final state of solid-body rotation is crucially dependent on the obstacle position. As found in previous studies, in the absence of any obstacle the relative flow becomes organized in a domain-filling regular cellular pattern, upon which it decays according to the well-known spin-up mechanism provided by the Ekman layer at the tank bottom. Since the obstacle acts both as a barrier and as a source of viscously produced wall vorticity, the formation of the cell pattern is in most cases drastically influenced (either impeded or promoted) by the solid obstacle. Theoretical predictions of the flow pattern in the starting stage agree very well with the laboratory observations.