  We demonstrate efficient transverse compression of a 12.5 MeV/c muon beam
stopped in a helium gas target featuring a vertical density gradient and
crossed electric and magnetic fields. The muon stop distribution extending
vertically over 14 mm was reduced to a 0.25 mm size (RMS) within 3.5 $\mu$s.
The simulation including cross sections for low-energy $\mu^+$-$\text{He}$
elastic and charge exchange ($\mu^+\leftrightarrow $ muonium) collisions
describes the measurements well. By combining the transverse compression stage
with a previously demonstrated longitudinal compression stage, we can improve
the phase space density of a $\mu^+ $ beam by a factor of $ 10^{10} $ with $
10^{-3} $ efficiency.
