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Let E be a separable Banach space with the λ-bounded approximation property. We show that for each ϵ > 0 there is a Banach space F with a Schauder basis such that E is isometrically isomorphic to a 1-complemented subspace of F and, moreover, the sequence (Tₙ) of canonical projections in F has the properties $su{p}_{n\in \mathbb{N}}\left|\right|Tₙ\left|\right|\le \lambda +ϵ$ and $limsu{p}_{n\to \infty }\left|\right|Tₙ\left|\right|\le \lambda$. This is a sharp quantitative version of a classical result obtained independently by Pełczyński and by Johnson, Rosenthal and Zippin.