It has been explained many times in the North America section.
You have two choices. You can get a 61.75 private "based on" your SA license. That is just a paper shuffle, no cost. All the limititions on your SA license apply to your FAA certificate and it is only valid while your SA license is.
Or you can pass the FAA written, get a FAA medical, fill whatever squares that have not been covered in training and pass a FAA checkride. Then you will have a "stand alone" certificate.