We present IRAM Plateau de Bure Interferometer observations of the CO ( 3–2 ) and CO ( 5–4 ) line transitions from a Ly \alpha blob at z \sim 2.7 in order to investigate the gas kinematics , determine the location of the dominant energy source , and study the physical conditions of the molecular gas . CO line and dust continuum emission are detected at the location of a strong MIPS source that is offset by \sim 1.5″ from the Ly \alpha peak . Neither of these emission components is resolved with the 1.7″ beam , showing that the gas and dust are confined to within \sim 7 kpc from this galaxy . No millimeter source is found at the location of the Ly \alpha peak , ruling out a central compact source of star formation as the power source for the Ly \alpha emission . Combined with a spatially-resolved spectrum of Ly \alpha and He ii , we constrain the kinematics of the extended gas using the CO emission as a tracer of the systemic redshift . Near the MIPS source , the Ly \alpha profile is symmetric and its line center agrees with that of CO line , implying that there are no significant bulk flows and that the photo-ionization from the MIPS source might be the dominant source of the Ly \alpha emission . In the region near the Ly \alpha peak , the gas is slowly receding ( \sim 100 km s ^ { -1 } ) with respect to the MIPS source , thus making the hyper-/superwind hypothesis unlikely . We find a sub-thermal line ratio between two CO transitions , I _ { CO ( 5 - 4 ) } / I _ { CO ( 3 - 2 ) } = 0.97 \pm 0.21 . This line ratio is lower than the average values found in high- z SMGs and QSOs , but consistent with the value found in the Galactic center , suggesting that there is a large reservoir of low-density molecular gas that is spread over the MIPS source and its vicinity .