Step 1: Understanding the Concept:
The question asks about the correct procedure and authority for an advocate who wants to transfer their enrollment from one State Bar Council's roll to another. This procedure is governed by the Advocates Act, 1961.
Step 2: Detailed Explanation:
Section 18 of the Advocates Act, 1961, deals with the "Transfer of name from one State roll to another".
It states that any person whose name is entered as an advocate on the roll of any State Bar Council may make an application in the prescribed form to the Bar Council of India for the transfer of his name from the roll of that State Bar Council to the roll of any other State Bar Council.
On receipt of the application, the Bar Council of India directs that the name of such person be removed from the roll of the first State Bar Council and entered in the roll of the other State Bar Council. The respective State Bar Councils must then comply with this direction.
Therefore, the central authority for approving such transfers is the Bar Council of India.
Step 3: Final Answer:
An application for transferring an advocate's enrollment from one state to another must be made to the Bar Council of India. Therefore, option (A) is the correct answer.