name acquired in connection with a previous marriage

This seems to be used in official translations into English:

Name after marriage
Following the marriage, both parties retain their own surname,
cf. Section 4, subs. 1 of Danish Act No. 193 of 29.4.1981 on Personal Names. However, either party can give notice that after the marriage they wish to assume the other party’s surname as their married name. The other party must consent to this. However, it is not possible to assume a surname which the other party has acquired by virtue of a previous marriage or registered partnership.

If a spouse has assumed the surname of the other spouse as married name, that person may use his or her own surname as a middle name preceding the married name. The marrying authority must be notified of this.

Own surname is the surname conferred upon the person at the time of birth or subsequently acquired on some basis other than marriage/registered partnership.

http://www.ambtokyo.um.dk/NR/rdonlyres/A63359EE-8FC6-46A6-B4...

http://www.gkchicago.um.dk/en/menu/Consular Services/Marriag...
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The verb bar is, as you say, the past tense of bære.

In fact a giftenavn like this is NOT an option for use after the (new) marriage - so I hope I have understood your question correctly.
But someone who has previously been married and divorced might still be using his/her married name.


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Note added at 4 hrs (2010-06-07 10:00:14 GMT)
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Of course, a person would be even more likely to be still using a previous married name after being widowed.