This paper designs an integrated estimator and ℒ1 adaptive control scheme to address the two main challenges involved in the Managed Pressure Drilling (MPD) system: first, the bottomhole states are updated at a low rate, which can be viewed as unmeasured and thus need to be estimated in real time; and second, the drilling process is subject to uncertainties including unknown parameters (e.g., frictions, densities), unmodeled actuator dynamics and noise, which require a robust adaptive controller for control of the bottomhole pressure. The estimator provides fast estimation of the bottomhole pressure and flow rate, based on the available measurements from the top-side. The ℒ1 adaptive controller drives the bottomhole pressure to the desired value following a reference model. We also provide a solution to handle the input delay. The design is based on a recently developed nonlinear drilling model. The results demonstrate that the ℒ1 adaptive controller has guaranteed performance bounds for both the input and the output signals of the system while using the estimation of the regulated outputs. Simulations that include different operational conditions verify the theoretical findings.