He can take a paternity test on his own. However, he would need her permission to test the child and, hopefully have herself tested. There are many firms and prices. However, you would want a lab that is AABB accredited.
One firm is Identigene.They do very accurate work. The Identigene kit is available in drug stores or online. The lab fee includes the child, alleged father and optionally, the mother.
This type of test is pretty good for the price, but sometimes the results are ambiguous. In that case or, if interested in the other parts of it, he might want to take one of the more expensive autosomal SNP tests. The best one to take would be:
Its Relative Finder will tell whether they are father and child. In addition, if the child is a boy, it tests his Y-haplogroup. If they are not in the same Y-haplogroup, your partner is not his father. It also tests for medically-related sites.
If you are not interested in the health aspects, he could take FTDNA's Family finder Test. However, it does not give the male haplogroup.