It's not illegal for a fiance to adopt your child but it is unusual and many judges frown on it. If you were to be married then the judge wouldn't object, assuming that there is nothing in your fiance's background (ex. a criminal history) that would interfere.
The judges that I've seen reject these applications have stated that if the fiance isn't committed enough to marry the other parent then they have doubts as to whether they are committed enough to adopt a child. This kind of thinking is old fashioned and eventually it will be rare to see a judge who thinks this way but it's not rare right now.
However, your lawyer can tell you which judges are likely to look at it that way and which ones are not and, again, if you get married then that removes the issue altogether.
So, the short answer is that it is legal for your fiance to adopt your child.
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