The null distributed controllability of the semilinear heat equation∂ty− Δy+g(y) =f1ωassuming thatg∈C1(ℝ) satisfies the growth condition lim sup|r|→∞g(r)∕(|r|ln3∕2|r|) = 0 has been obtained by Fernández-Cara and Zuazua (2000). The proof based on a non constructive fixed point theorem makes use of precise estimates of the observability constant for a linearized heat equation. Assuming thatg′ is bounded and uniformly Hölder continuous on ℝ with exponentp∈ (0, 1], we design a constructive proof yielding an explicit sequence converging strongly to a controlled solution for the semilinear equation, at least with order 1 +pafter a finite number of iterations. The method is based on a least-squares approach and coincides with a globally convergent damped Newton method: it guarantees the convergence whatever be the initial element of the sequence. Numerical experiments in the one dimensional setting illustrate our analysis.