We consider the norm closure Θ of the algebra of all operators of order and class zero in Beutet de Monvel's calculus on a manifold X with boundary ∂X. We first describe the image and the kernel of the continuous extension of the boundary principal symbol homomorphism to Θ. If X is connected and ∂X is not empty, we then show that the K-groups of Θ are topologically determined. In case ∂X has torsion free K-theory, we get K_i(Θ/ℜ) ≃ K_i(C(X)) ⊕ K ₁-_i(C₀(T*Ẋ)), i = 0, 1, with ℜ denoting the compact ideal, and T*Ẋ denoting the cotangent bundle of the interior. Using Boutet de Monvel's index theorem, we also prove that the above formula holds for i = 1 even without this torsion-free hypothesis; and show, moreover, that K₁ (Θ) ≃ K₁ (C(X)) ⊕ ker χ, with χ: K₀(T*Ẋ) → Z denoting the topological index. For the case of orientable, two-dimensional X, K ₀(Θ) ≃ ℤ^2g+m and K₁(Θ) ≃ ℤ^2g+m-1, where g is the genus of X and m is the number of connected components of ∂X. We also obtain a composition sequence 0 ⊂ ℜ ⊂ script H sign ⊂ Θ, with Θ/script H sign commutative and script H sign/ℜ isomorphic to the algebra of all continuous functions on the cosphere bundle of ∂X with values in compact operators on L²(ℝ̄₊).