Final answer:
The process for appointing a new board member in a U.S. Congressional context depends on whether it pertains to the House of Representatives or other federal positions. For a House vacancy, the governor of the state may call for a special election. For federal appointments such as judges, the President nominates a candidate with the Senate's approval.
Step-by-step explanation:
In the context of a U.S. Congressional board, where a board member is disqualified and the remaining members do not constitute a majority, the process for appointing a new member varies depending on the specific circumstances.
Article I, Section 2 of the U.S. Constitution outlines that when a House member dies or resigns during the term, the governor of the state from where the representative is may call for a special election to fill the vacancy. In contrast, as per Article II, Section 2, appointments for certain federal positions, like federal judges, are made by the President with the advice and consent of the Senate.
If the question pertains to local or state boards, the process might be different and could be governed by state law or specific board rules. For example, a local board of zoning appeals may have its own rules or rely on state statutes for filling a vacancy on the board.