The priest must wear the chasuble. Redemptionis Sacramentum states:
“The vestment proper to the Priest celebrant at Mass, and in other sacred actions directly connected with Mass unless otherwise indicated, is the chasuble, worn over the alb and stole. Likewise the Priest, in putting on the chasuble according to the rubrics, is not to omit the stole. All Ordinaries should be vigilant in order that all usage to the contrary be eradicated” (123).
The only times when a chasuble may not be worn by a priest celebrating Mass would be if he were concelebrating a Mass and were not the principal celebrant and it were absolutely necessary that this be done “for a just reason” (Redemptionis Sacramentum 124). Other conceivable reasons not to wear a chasuble would be when Mass is celebrated in extremely hot conditions (thus, to prevent heat stroke) or when no chasuble is available like in a military Mass.
Alternatively, the cincture serves a more strictly functional purpose. Redemptionis Sacramentum states:
“The alb” is “to be tied at the waist with a cincture unless it is made so as to fit even without a cincture" (212).