From: https://law.justia.com/codes/rhode-island/2019/title-17/chapter-17-4/section-17-4-9/
§ 17-4-9. Special election to fill senatorial vacancy.
Whenever a person elected a senator in congress, at any time between the day of that person's election and the beginning of his or her term of office, refuses to serve and so declares to the secretary of state, or dies, becomes insane, removes from the state, or is otherwise incapacitated, or whenever a vacancy happens in the representation of this state in the United States senate, the governor shall issue his or her writ of election directed to the several city and town clerks, or local boards as the case may be, ordering a new election of senator to fill the vacancy to be held in the state at as early a date, to be stated in the writ, as will be in compliance with the provisions of law in relation to these elections, but no election provided for by this section shall be held on any day other than the first Tuesday next after the first Monday of any month; provided, that whenever a vacancy occurs between the first day of July and the first day of October in any even-numbered year, the governor shall, unless in his or her opinion the public good requires an earlier special election, issue his or her writ for a special election to fill the vacancy to be held with the general election on the Tuesday next after the first Monday in November of that year.
It seems rather unlikely a sitting Democratic senator will be chosen for Biden's cabinet with the possibility of the Senate being split 50-50 in January.
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