  We study the stability of the kaon condensed color-flavor locked (CFL) phase
of dense quark matter with regard to the formation of a non-zero Goldstone
boson current. In the kaon condensed phase there is an electrically charged
fermion which becomes gapless near \mu_s^(1) \simeq 1.35\Delta and a neutral
fermion which becomes gapless near \mu_s^(2)\simeq 1.61\Delta. Here,
\mu_s=m_s^2/(2p_F) is the shift in the Fermi energy due to the strange quark
mass m_s and \Delta is the gap in the chiral limit. The transition to the
gapless phase is continuous at \mu_s^(1) and first order at \mu_s^(2). We find
that the magnetic screening masses are real in the regime \mu_s< \mu_s^(2), but
some screening masses are imaginary for \mu_s> \mu_s^(2). We show that there is
a very weak current instability for \mu_s>\mu_s^(1) and a more robust
instability in a small window near \mu_s^(2). We also show that in the
Goldstone boson current phase all components of the magnetic screening mass are
real. There is a range of values of \mu_s below 2\Delta in which the magnetic
gluon screening masses are imaginary but the phase is stable with respect to
electrically neutral fluctuations of the gauge field.
