If I have a K1 Visa (the fiancee visa), can my Danish fiance then obtain permanent residence in the United States if we marry in Denmark?

No

If you get married in Denmark you need a spousal visa

If you have already applied for or been approved for a fiasnce visa you get married in the US

It doesn't matter where you marry as long as the sponsoring US citizen partner has a US income high enough to support the petition and the foreign partner doesn't have a criminal record.

From your comment to Brother Hesekiel's answer, you have not started the process of getting a fiance visa. If that is correct and you are certain the relationship is real then the best option for you is to marry and then you sponsor them for a spouse visa.

If the two of you marry in Europe then get the marriage license translated into English by a translator approved by the US Embassy.

If they travel to the USA on the visa waiver program (ESTA) and the two of you marry then your fiance needs to say NOTHING to the US Immigration Officer about getting married to a US citizen when they enter the USA. They are "visiting a friend", which is true until the two of you actually get married. Your fiance also MUST exit the USA on schedule and return home to wait while you file to sponsor them for the spouse visa.

https://www.uscis.gov/family/family-us-citizens/br...

Note that > you < must have a steady job making at last $22,000 per year with three years of income tax returns to prove it to qualify to sponsor your spouse for immigration. Same income requirement for a fiance/e visa.

No, don't be silly. The K-1 fiancé visa allows your foreign fiancé into the US to get married to you in US. Only if you two marry in the US within 90 days of arrival, then file for Adjustment of Status, can your new spouse remain in US & get a temporary (conditional) green card.

If already married, the K-1 fiancé visa becomes invalid. The US-spouse must start from scratch, apply for a spouse (CR-1) visa for the new foreign spouse to enter US. Spouse visas typically take longer to obtain than fiancé visas, and can take twice as long. Your foreign spouse cannot enter US until approved for & receives the spouse visa.

Why invalidate the fiancé visa your fiancé already has, and start over again? Waste of time & money.

Your post is confusing to me.

The K-1 is a visa for a foreigner, i.e., a Danish national, to travel to the United States with the purpose of getting married to their US citizen fiance. That has to happen within 90 days upon arrival.

If you were to get married in Denmark, your fiance would no longer be your fiance but your spouse and the visa becomes invalid after the fact. You then would have to file an I-130 petition leading to a spousal visa, the CR-1, about a year later.

I'm of course not sure if I understood your question correctly, but I would have never even file an I-129F for a K-1. Why? Why didn't you just travel to Denmark, get married, then file the I-130 instead? Much, much cheaper as no Adjustment of Status is needed.