In this paper we prove that the proper Dirichlet subalgebras of the disc algebra discovered by Browder and Wermer [1] are maximal subalgebras of the disc algebra (Theorem 2). We also give an extension to general function algebras of a theorem of Rudin [4] on the existence of maximal subalgebras of C(X). Theorem 1 implies that every function algebra defined on an uncountable metric space has a maximal subalgebra.A function algebra A on X is a uniformly closed, point-separating subalgebra of C(X), containing the constants, where X is a compact Hausdorff space. If A and B are function algebras on X, A ⊂ B, A ≠ B, we say A is a maximal subalgebra of B if whenever C is a function algebra on X with A ⊂ C ⊂ B, either C = A or C = B.