We present IRAM Plateau de Bure Interferometer 3 mm observations of CO J ( 1 \rightarrow 0 ) emission in two 24 \mu m-selected starburst galaxies in the outskirts ( \sim 2–3 R _ { virial } ) of the rich cluster Cl 0024+16 ( z = 0.395 ) . The galaxies ’ inferred far-infrared luminosities place them in the luminous infrared galaxy class ( LIRGs , L _ { FIR } > 10 ^ { 11 } L _ { \odot } ) , with star formation rates of \sim 60 M _ { \odot } yr ^ { -1 } . Strong CO J ( 1 \rightarrow 0 ) emission is detected in both galaxies , and we use the CO line luminosity to estimate the mass of cold molecular gas , M ( { H _ { 2 } } ) . Assuming M ( { H _ { 2 } } ) / L ^ { \prime } _ { CO } = 0.8 M _ { \odot } ( { K km ^ { -1 } pc ^ { 2 } } ) ^ { -1 } , we estimate M ( { H _ { 2 } } ) = ( 5.4 – 9.1 ) \times 10 ^ { 9 } M _ { \odot } for the two galaxies . We estimate the galaxies ’ dynamical masses from their CO line-widths , M _ { dyn } \sim 1 – 3 \times 10 ^ { 10 } M _ { \odot } , implying large cold gas fractions in the galaxies ’ central regions . At their current rates they will complete the assembly of M _ { \star } \sim 10 ^ { 10 } M _ { \odot } and double their stellar mass within as little as \sim 150 Myr . If these galaxies are destined to evolve into S0s , then the short time-scale for stellar mass assembly implies that their major episode of bulge growth occurs while they are still in the cluster outskirts , long before they reach the core regions . Subsequent fading of the disc component relative to the stellar bulge after the gas reservoirs have been exhausted could complete the transformation of spiral-to-S0 .