A known method to compute on an asynchronously updating cellular automaton is the simulation of a synchronous computing model on it. Such a scheme requires not only an increased number of cell states, but also the simulation of a global synchronization mechanism. Asynchronous systems tend to use synchronization only on a local scale—if they use it at all. Research on cellular automata that are truly asynchronous has been limited mostly to trivial phenomena, leaving issues such as computation unexplored. This paper presents an asynchronously updating cellular automaton that conducts computation without relying on a simulated global synchronization mechanism. The two-dimensional cellular automaton employs a Moore neighborhood and 85 totalistic transition rules describing the asynchronous interactions between the cells. Despite the probabilistic nature of asynchronous updating, the outcome of the dynamics is deterministic. This is achieved by simulating delay-insensitive circuits on it, a type of asynchronous circuit that is known for its robustness to variations in the timing of signals. We implement three primitive operators on the cellular automaton from which any arbitrary delay-insensitive circuit can be constructed and show how to connect the operators such that collisions of crossing signals are avoided.