Synchronized oscillators are ubiquitous in nature1, and synchronization plays a key part in various classical and quantum phenomena. Several experiments2, 3, 4 have shown that in thin superconducting films, disorder enforces the droplet-like electronic texture—superconducting islands immersed into a normal matrix—and that tuning disorder drives the system from superconducting to insulating behaviour. In the vicinity of the transition, a distinct state4 forms: a Cooper-pair insulator, with thermally activated conductivity. It results from synchronization of the phase of the superconducting order parameter at the islands across the whole system5. Here we show that at a certain finite temperature, a Cooper-pair insulator undergoes a transition to a superinsulating state with infinite resistance. We present experimental evidence of this transition in titanium nitride films and show that the superinsulating state is dual to the superconducting state: it is destroyed by a sufficiently strong critical magnetic field, and breaks down at some critical voltage that is analogous to the critical current in superconductors.