  We have used the Green Bank Telescope to carry out a deep search for
redshifted CO J=2-1 line emission from an extended (>17 kpc) Ly-Alpha blob
(LAB), "Himiko", at z~6.595. Our non-detection of CO J=2-1 emission places the
strong 3-sigma upper limit of L'_CO < 1.8x10^10 x sqrt(dV/250) K km/s pc^2 on
the CO line luminosity. This is comparable to the best current limits on the CO
line luminosity in LABs at z~3 and lower-luminosity Lyman-Alpha emitters (LAEs)
at z>~6.5. High-z LABs appear to have lower CO line luminosities than the host
galaxies of luminous quasars and sub-mm galaxies at similar redshifts, despite
their high stellar mass. Although the CO-to-H2 conversion factor is uncertain
for galaxies in the early Universe, we assume X_CO = 0.8 Msun (K km/s pc^2)^-1
to obtain the limit M(H_2) < 1.4 x 10^10 Msun on Himiko's molecular gas mass;
this is a factor of >2.5 lower than the stellar mass in the z~6.595 LAB.
