If X is a stable process of index α∈(0, 2) whose Lévy measure has density cx^−α−1 on (0, ∞), and S₁=sup_0<t≤1X_t, it is known that P(S₁>x)∽Aα⁻¹x^−α as x→∞ and P(S₁≤x)∽Bα⁻¹ρ⁻¹x^αρ as x↓0. [Here ρ=P(X₁>0) and A and B are known constants.] It is also known that S₁ has a continuous density, m say. The main point of this note is to show that m(x)∽Ax^−(α+1) as x→∞ and m(x)∽Bx^αρ−1 as x↓0. Similar results are obtained for related densities.