  There exist consistent low energy effective field theories describing gravity
in the Higgs phase that allow the coexistence of massive gravitons and the
conventional 1/r potential of gravity. In an effort to constrain the value of
the graviton mass in these theories, we study the tensor contribution to the
CMB temperature anisotropy and polarization spectra in the presence of a
non-vanishing graviton mass. We find that the observation of a B-mode signal
consistent with the spectrum predicted by inflationary models would provide the
strongest limit yet on the mass of an elementary particle -- a graviton -- at a
level of m\lesssim 10^(-30) eV\approx(10 Mpc)^(-1). We also find that a
graviton mass in the range between (10 Mpc)^(-1) and (10 kpc)^(-1) leads to
interesting modifications of the polarization spectrum. The characteristic
signature of a graviton mass in this range would be a plateau in the B-mode
spectrum up to angular multipoles of l\sim 100. For even larger values of the
graviton mass the tensor contribution to the CMB spectra becomes strongly
suppressed.
