In significantly underexpanded jets, screech inherently ceases to exist. This paper studies screech cessation in a supersonic rectangular jet and provides an explanation for its occurrence. Experimental data are presented for fully expanded Mach numbers, Mj, ranging from 1.1 to 1.9. Screech becomes unsteady beyond Mj=1.65 and ceases to exist beyond Mj=1.75. The reason for this cessation has remained a mystery, and this paper examines three suspects: (i) the theory of a frequency mismatch between screech tones and the band of the most-amplified jet instability waves, (ii) the notion that Mach disk formation disrupts the shock-cell structure and weakens the screech-producing shocks, and (iii) the idea that acoustic feedback and receptivity diminish at high levels of underexpansion. A thorough interrogation of experimental data shows that (i) is not the main cause of screech cessation here, (ii) plays an insignificant role, and (iii) appears to have been largely responsible for screech cessation. Cessation occurs because feedback to the jet lip is diminished due to excessive expansion of the jet boundary. Further, since the jet lip now reflects and scatters low intensity sound, the end result is poor receptivity at the initial shear layer. This theory is substantiated by the re-activation of screech when the nozzle lip thickness is made larger than the expanded jet boundary. Finally, increasing lip thickness is seen to produce a systematic shift (to higher Mj) of the onset of cessation. The results of this study are of direct relevance to the sonic fatigue problem in aircraft structures, because understanding screech helps prevent such damage.