The status of the Cabinet appears to be uncertain in that case. The Legislative Assembly and Executive Council Act states that the Executive Council, also known as Cabinet, is responsible for the overall direction and management of the GNWT. The act also provides that the members of the Executive Council are to be appointed upon the recommendation of the Legislative Assembly and hold office "during the pleasure" of the Assembly. Finally, the act provides that the Premier shall continue to hold office following dissolution of the Legislative Assembly until the next Legislative Assembly chooses the next Premier.
Currently, Ministers continue in office between Assemblies until an election is held, a new Assembly is convened, and a new Cabinet is chosen. This is generally similar to practice in all federal, provincial and territorial governments in Canada. What is unclear is whether or not this practice could continue if the Assembly were dissolved without any arrangements for the election of a successor.