  A first flavour-tagged measurement of the time-dependent CP-violating
asymmetry in $B_s^0 \to \phi\phi$ decays is presented. In this decay channel,
the CP-violating weak phase arises due to CP violation in the interference
between $B_s^0$-$\bar{B}_s^0$ mixing and the $b \to s \bar{s} s $ gluonic
penguin decay amplitude. Using a sample of $pp$ collision data corresponding to
an integrated luminosity of $1.0\; fb^{-1}$ and collected at a centre-of-mass
energy of $7 \rm TeV$ with the LHCb detector, $880\ \B_s^0 \to \phi\phi$ signal
decays are obtained. The CP-violating phase is measured to be in the interval
[-2.46, -0.76] \rm rad$ at 68% confidence level. The p-value of the Standard
Model prediction is 16%.
