No, a monopolist does not achieve efficiency in its production. Instead, it produces too little of its product and sells it at too high of a price. This means that a market that is controlled by a monopolist can never be efficient.
Like any firm, a monopolist produces at the quantity where marginal revenue equals marginal cost. However, in a monopoly, this is not the same as the quantity where price equals marginal cost. Instead, it is a lower quantity because the marginal revenue curve is downward sloping. With the lower quantity, the price is also higher than it should be.
Therefore, a monopolist does not achieve efficiency because it produces too little and charges too much for what it sells.