  Vacuum birefringence (VB) is a basic phenomenon predicted in quantum
electrodynamics (QED). However, due to the smallness of the signal,
conventional magnet-based and extremely intense laser-driven detection methods
are still very challenging. This is because in the first case the interaction
length is large but the field is limited, and vice versa in the second case. We
put forward a method to generate and detect VB in a plasma bubble wakefield,
which combines both advantages, providing large fields along large interaction
lengths. A polarized $\gamma$-photon beam is considered to probe the wakefield
along a propagation distance of millimeters to centimeters in the plasma
bubble. We find via plasma particle-in-cell simulations that the VB signal in
terms of Stokes parameters can reach about $ 10^{-5}$ ($10^{-3}$-$10^{-2}$) for
tens of MeV (GeV) probe photons with moderately intense lasers
($10^{20}$-$10^{21}~\mathrm{W/cm^2}$). The main source of noise from plasma
electrons is mitigated, in particular, by a choice of $\gamma$-photon
polarization and by proper modulation of the plasma density. The proposed
method represents an attractive alternative for the experimental observation of
VB via laser-plasma interaction.
