In order to constrain the bolometric luminosities , dust properties and molecular gas content of giant Ly \alpha nebulae , the so-called Ly \alpha blobs , we have carried out a study of dust continuum and CO line emission in two well-studied representatives of this population at z \sim 3 : a Ly \alpha blob discovered by its strong Spitzer MIPS 24 \mu m detection ( LABd05 ; Dey et al . 2005 ) and the Steidel blob 1 ( SSA22-LAB01 ; Steidel et al . 2000 ) . We find that the spectral energy distribution of LABd05 is well described by an AGN-starburst composite template with L _ { FIR } = ( 4.0 \pm 0.5 ) \times 10 ^ { 12 } L _ { \sun } , comparable to high- z sub-millimeter galaxies and ultraluminous infrared galaxies . New Large APEX Bolometer Camera ( LABOCA ) 870 \mu m measurements rule out the reported Submillimeter Common-User Bolometer Array ( SCUBA ) detection of the SSA22-LAB01 ( S _ { 850 \mu { m } } = 16.8 mJy ) at the > 4 \sigma level . Consistent with this , ultra-deep Plateau de Bure Interferometer ( PdBI ) observations with \sim 2″ spatial resolution also fail to detect any 1.2 mm continuum source down to \approx 0.45 mJy beam ^ { -1 } ( 3 \sigma ) . Combined with the existing ( sub ) mm observations in the literature , we conclude that the FIR luminosity of SSA22-LAB01 remains uncertain . No CO line is detected in either case down to integrated flux limits of S _ { \nu } \Delta V \lesssim 0.25 – 1.0 Jy km s ^ { -1 } , indicating a modest molecular gas reservoir , M ( { H _ { 2 } } ) < 1–3 \times 10 ^ { 10 } M _ { \sun } . The non-detections exclude , with high significance ( 12 \sigma ) , the previous tentative detection of a CO J = 4–3 line in the SSA22-LAB01 . The increased sensitivity afforded by the Atacama Large Millimeter/submillimeter Array will be critical in studying molecular gas and dust in these interesting systems .