This paper proposes a regularization method for determining the Caputo derivative of order 𝛼 ∈ (1, 2) from noisy data in 𝐿∞[0, 1]. The exact data function 𝑞 ∈ 𝐶4[0, 1] is assumed to satisfy 𝑞(0) = 𝑞′(0) = 𝑞″(0) = 𝑞‴(0) = 0 and ‖𝑞(4)‖𝐿∞[0,1] ≤ 𝐸, where 𝐸 > 0. The method approximates the second derivative of 𝑞 by the solution of a well-posed problem depending on a regularization parameter 𝛽 > 0, and then uses this approximation to construct an approximation of the Caputo derivative. Under a priori parameter choice rules, Hölder-type error estimates of order 𝛿1/4 are established as 𝛿 → 0, where 𝛿 denotes the noise level measured in the 𝐿∞[0, 1] norm.