  We study the entanglement between a qubit and its environment from the
spin-boson model with Ohmic dissipation. Through a mapping to the anisotropic
Kondo model, we derive the entropy of entanglement of the spin
$E(\alpha,\Delta,h)$, where $\alpha$ is the dissipation strength, $\Delta$ is
the tunneling amplitude between qubit states, and $h$ is the level asymmetry.
For $1-\alpha \gg \Delta/\omega_c$ and $(\Delta,h) \ll \omega_c$, we show that
the Kondo energy scale $T_K$ controls the entanglement between the qubit and
the bosonic environment ($\omega_c$ is a high-energy cutoff). For $h\ll T_K$,
the disentanglement proceeds as $(h/T_K)^2$; for $h\gg T_K$, $E$ vanishes as
$(T_K/h)^{2-2\alpha}$, up to a logarithmic correction. For a given $h$, the
maximum entanglement occurs at a value of $\alpha$ which lies in the crossover
regime $h\sim T_K$. We emphasize the possibility of measuring this entanglement
using charge qubits subject to electromagnetic noise.
