Even though the conditions dont mention it, France CANT exist either as a free nation or as a puppet or else the achievement wont pop. I tried to do this 3 times each time making France a puppet and it never popped.
Here are the conditions:
1) Royal Marriage with King Edward
2) Germany MUST be Axis. Its best you select "historic focus" to avoid the chance of Germany taking the "oppose Hitler" path.
3) You must control Paris by annexing the region.
4) France CANT exist as a puppet. Im unsure if Vichy France can exist (someone confirm) so to be safe just try to annex all of mainland France if you can.
TIPS:
1) Build up your army quick. I suggest the standard 20W (7 inf 2 artillery) with field hospital support upgrade. About 80-100 of these divisions should do.
2) Research the second level of the amphibious invasion tech. You need it as sending 10 divisions is suicide. I sent my entire force around Dunkirk (near Belgium). The French navy in the Pacific is very weak and you should have no problem gaining Naval superiority especially if you trap their Mediterranean fleet at Gibraltar and Suez.
3) Once your troops land make a very aggressive push for Paris and some nearby major cities. France has very low stability and war support and capturing paris and 2 or 3 nearby towns is usually enough to capitulate them.