Models of insurance with adverse selection predict that only the best risks— those least likely to suffer a loss—are uninsured, a prediction at odds with coverage denials for pre-existing conditions. They also typically assume that insurance provision is costless: the only cost is claims payment. We introduce costly insurance provision into a standard monopoly insurance model with adverse selection. We show that with loading or a fixed cost of claims processing, the insurer denies coverage only to those likely to be the worst risks. Unexpectedly, it turns out that loading also overturns three classic textbook properties of monopoly pricing models: no one is pooled with the highest consumer type; the highest type gets an efficient contract; and all other types get contracts distorted downwards from their efficient contracts. Indeed all types can be pooled on a single contract. Finally we show that both loading and a fixed claims cost do not affect qualitative properties of (Rothschild-Stiglitz) competitive equilibrium (when it exists), so these costs generate potentially testable implications of competitive vs monopoly insurance: for example, no competitive equilibrium can be pooling.