  We present results from a sensitive search for CO J=1-0 line emission in two
z> 6.5 Lyman-alpha emitters (LAEs) with the Green Bank Telescope. CO J=1-0
emission was not detected from either object. For HCM 6A, at z ~ 6.56, the
lensing magnification factor of ~4.5 implies that the CO non-detection yields
stringent constraints on the CO J=1-0 line luminosity and molecular gas mass of
the LAE, L'(CO) < 6.1x10^9 x (dV/300)^(1/2) K km/s pc^2 and M(H_2) < 4.9x10^9 x
(dV/300)^(1/2) x (X(CO)/0.8) Msun. These are the strongest limits obtained so
far for a z >~ 6 galaxy. For IOK-1, the constraints are somewhat less
sensitive, L'(CO) < 2.3x10^10 x (dV/300)^(1/2) K km/s pc^2 and M(H_2) <
1.9x10^10 x (dV/300)^(1/2) x (X(CO)/0.8) Msun. The non-detection of CO J=1-0
emission in HCM~6A, whose high estimated star formation rate, dust extinction,
and lensing magnification make it one of the best high-z LAEs for such a
search, implies that typical z >~ 6 LAEs are likely to have significantly lower
CO line luminosities than massive sub-mm galaxies and hyperluminous infrared
quasars at similar redshifts, due to either a significantly lower molecular gas
content or a higher CO-to-H_2 conversion factor.
