The Arabic in that hadith literally just says "…والروم…" ("…and Rome…") It doesn't say, or even imply, anything about conquering, much less being conquered by Muslims.
The sign that it's likely referring to is found in the Quran:
 Alif, Lam, Meem.
 The Byzantines (Rome) have been defeated
 In the nearest land. But they, after ...
The closest wording in a hadith to what you've mentioned appears in:
'Abdullah said, 'From the Prophet, may Allah bless him and grant him peace, who said, "Before the Final Hour people will single out one individual for the greeting, commerce will increase until a woman helps her husband in business, people will sever their links with their relatives, ...