Sources close to the couple tell TMZ ... the Migos rapper woke up one September morning, looked at the "Bodak Yellow" rapper in the eye and straight-up said, "Marry me." She obviously said yes but, believe it or not, that impulse was only the beginning.

We're told, that same day Offset and Cardi -- who at this point, we're told was not pregnant -- hired an officiant to come to the house and marry them in their bedroom. The only person besides the officiant to witness the nuptials was Cardi's cousin.

It was such a spur-of-the-moment decision, there was obviously no time for an engagement ring, much less a wedding band. That brings us to the "public engagement" back in October. We're told that was NOT Offset proposing to Cardi. It was Offset keeping his promise to Cardi and getting her a ring. As we first reported ... at that point they had been married for at least a month.

As for why Offset ain't flaunting a wedding band ... our sources say he'll get one and wear it once there's an official wedding ceremony, which we're told will happen after Cardi pops.