You are right - King Michael did not approve, and Sophie thus lost her title. This is a sample of what can be found on the internet - for what it is worth. Probably it is more or less correct:
1)http://www.geocities.com/dagtho/uk-succession.html
15. Sophie de Laufenborg, née Princess of Romania, married at Neuilly 29 August 1998 Alain Michel Léonce Biarneix (b. Nancy 10 July 1957). The marriage was not approved by HM The King of Romania, whereupon Mrs. de Laufenborg lost her style, title and rank as Princess of Romania. Her husband, who calls himself "de Laufenborg", converted to the Orthodox faith prior to the wedding, which means that Mrs. de Laufenborg and her daughter are in line of succession to the British throne. The couple have now divorced, and the former Mrs. de Laufenborg has been reconciled with her parents.
2)http://www.btinternet.com/~allan_raymond/Roumanian_Royal_Family.htm
Princess Sophia lost her title and status on 29 August 1998 when she married without her father's permission. Alain Biarneix is also known as Michel de Laufenborg
3)http://freepages.genealogy.rootsweb.com/~byzantium/royaldata6.txt
Sophie of Romania, daughter of King Michael, is listed here: She m.Michel Biarneix de Laufenbourg & has issue:
(1) Elisabeta-Maria Laufenbourg, b.AUG 15 1999 in Paris. Sophie was stripped of her title and succession rights upon her marriage.
Biarneix sounds like a cad....
