A scheme is proposed for making highly rotationally excited diatomic molecules ("super rotors") in their ground vibrational and electronic state, e.g., (Li2X)-Li-6 (1)Sigma(g)(+) (v=0,J greater than or equal to 115) where the rotational energy exceeds the bond strength (E(0,J)-E(0,0)greater than or equal to D-0(0)). Such levels, while strictly speaking quasibound, have very long tunneling lifetimes (> 10(11) s for J less than or equal to 130), and should have very interesting and unique collisional properties, especially at low temperature. The rotation of the molecules is "spun up" by sequential irradiation by R branch photons in the A (1)Sigma(u)(+)-X (1)Sigma(g)(+) bands starting with cold molecules at low J. Spontaneous emission to other vibrational levels is overcome by using a pump laser and its multiple Raman sidebands as in previous work on "spinning down" diatomics. (C) 2000 American Institute of Physics. [S0021-9606(00)00914-4].