Two molecular cloudlets have been detected and mapped in a partial survey of 12CO emission in a diffuse H I cloud toward Perseus. Previous observations of K I absorption and 21 cm emission had shown that the cloud is within 120 pc of the Sun, with a diameter of 14 pc, a total mass 1500 Msun, and temperatures as low as 30 K, making it a good candidate to contain molecular gas. The two cloudlets embedded in the H I cloud have diameters ˜1 pc, molecular densities 100 cm-3, and molecular masses of only 1 Msun, and lie in front of the two stars with the strongest detected K I absorption. The derived total hydrogen column density is 1 x 1021 cm-2 The molecular cores detected are not virialized but confined by the turbulent pressure of the interstellar medium. No 12CO was detected along the lines of sight to eight stars showing K I absorption, or in the direction of the biggest concentration of H I.