The formal acknowledgement/approval by the Department of State (DOS) is manifested by the issuance of a Certificate of Loss of Nationality of the United States, Form DS-4083 (CLN). See sample form in this post.
For U.S. federal income tax purposes, the actual receipt of the CLN by the individual is necessary for numerous reasons too. See, The Importance of a Certificate of Loss of Nationality (“CLN”) and FATCA – Foreign Account Tax Compliance Act.
You should be aware that the wait times to receive the CLN can vary wildly depending upon where (which Embassy or Consulate office) and when one goes to take the oath of renunciation. I have seen some issued as soon as 30 days. More commonly, 4 months is a fairly common time frame; but I have also seen cases where it took more than 9 months.