This paper presents a non-asymptotic upper bound for the estimation error of the constrained lasso, under the high-dimensional ($n \ll p$) setting. In contrast to existing results, the error bound in this paper is sharp, is valid when the parameter to be estimated is not exactly sparse (e.g., when it is weakly sparse), and shows explicitly the effect of over-estimating the $\ell_1$-norm of the parameter to be estimated on the estimation performance. The results of this paper show that the constrained lasso is minimax optimal for estimating a parameter with bounded $\ell_1$-norm, and also for estimating a weakly sparse parameter if its $\ell_1$-norm is accessible.