We present sensitive CO ( J = 1 \rightarrow 0 ) emission line observations of three metal-poor dwarf irregular galaxies Leo P ( Z \sim 3 % Z _ { \odot } ) , Sextans A ( Z \sim 7.5 % Z _ { \odot } ) , and Sextans B ( Z \sim 7.5 % Z _ { \odot } ) , all obtained with the Combined Array for Millimeter-wave Astronomy ( CARMA ) interferometer . While no CO emission was detected , the proximity of the three systems allows us to place very stringent ( 4 \sigma ) upper limits on the CO luminosity ( L _ { CO } ) in these metal-poor galaxies . We find the CO luminosities to be L _ { CO } < 2900 K km/s pc ^ { 2 } for Leo P , L _ { CO } < 12400 K km s ^ { -1 } pc ^ { 2 } for Sextans A , and L _ { CO } < 9700 K km s ^ { -1 } pc ^ { 2 } for Sextans B . Comparison of our results with recent observational estimates of the factor for converting between L _ { CO } and the mass of molecular hydrogen , as well as theoretical models , provides further evidence that either the CO-to-H _ { 2 } conversion factor increases sharply as metallicity decreases , or that stars are forming in these three galaxies very efficiently , requiring little molecular hydrogen .