Let \(K \subset \mathbb{R}^ n\) be a compact polyhedron and let \(Y\) be an at least \(n\)-dimensional linear subspace of the Hilbert space \(\ell_ 2\). A map \(f:K \to Y\) is said to be \(L\)-bilipschitz if \(L \geq 1\) and \[ {| x-y | \over L} \leq | fx-fy | \leq L | x-y | \] for all \(x,y \in K\). Although simple examples show that in general an \(L\)- bilipschitz map \(f:K\to Y\) need not have an extension to any embedding \(g:\mathbb{R}^ n \to Y\), it is shown that even bilipschitz extensions exist if \(L\) is sufficiently close to 1. The exact result is as follows: Let \(K\) and \(Y\) be as above. Then there exists \(L_ 0>1\) and a function \(L_ 1: [1,L_ 0] \to [1,\infty[\) with \(L_ 1(1)=1=\lim_{L \to 1} L_ 1(L)\) and such that if \(1 \leq L \leq L_ 0\) and if \(f:K \to Y\) is an \(L\)-bilipschitz mapping, then \(f\) has an \(L_ 1(L)\)-bilipschitz extension \(g:\mathbb{R}^ n \to Y\).