I agree with your concerns for the girl. There are some great suggestions in the other answers, so I'll limit myself mostly to consequences for the girl. Let's assume you can't stop them from having sex, which doesn't mean the discussion shouldn't take place; they can still be in a relationship with each other that doesn’t involve sex.
Discussions with your son should always be welcome, honest, patient, and respectful, so that they can come to you with any problem they might have.
Your son states he is adamant that he loves her and only wants the best for her. At 15, and certainly at 13; they probably doesn't know what's best for her. Discuss with him what “best for her” means. That does mean taking special care not to hurt her (she needs love and respect from people she cares about). In adolescents, early sexual experiences play a significant part in their future ability to form solid, trusting relationships. While such sexual experiences can be positive, they can also have a harmful effect on mental and physical health and development. It is important that young people are mature and ready before they engage in sexual activity.
Young adolescents tend to be less deliberative and rational about sexual decisions than older persons. There is an ever-increasing sexualization of pre-adolescent girls in western societies. Is he aware of the damaging effects on decision-making that this has on girls? What is his definition of love and respect? (For that matter, what is his definition of "enthusiastic consent" and "mutually pleasurable"?)
Sex at 13 is illegal. Even adults often feel like doing lots of things, but can’t because the law says they can’t. Discuss the ramifications of this with your son. But worse that merely illegal, the female usually bears the brunt of the consequences of sex at this age. Her parents have a right to know what's going on with her. Is your son ready for what might happen to him and her when her parents find out? Who will tell the parents?
What will happen to her reputation? Sexually active girls are often treated differently by other young people. Terms such as "slut" and "slag" are thrown around with alarming ease. If they break up, how will having had sex affect her future relationships? (She is more likely to be coerced/manipulated into having sex by future partners.)
The younger adolescents are when they begin having sex, the greater their risks of negative consequences, and early sexual intercourse experiences often are psychologically coercive. For the girl, learning more about social pressures, negotiation, and refusal skills is important. Even the small difference in their ages for her may represent a power imbalance. If she is not enthusiastically consenting (she is only 13), they shouldn't be having sex.
Contraception: if he's not using contraception every time, then he's not doing what's best for her. Seven in ten pregnancies to adolescent teens occur to teens who were not using any method of contraception when they became pregnant. Adolescents engage in risky sexual behaviors because they believe that they have little to lose. Adolescents who value education and job success and who perceive their future opportunities optimistically should have stronger motivation for avoiding early pregnancy and parenthood.
Girls from abusive backgrounds and those with poor relationships with their parents tend to have sex earlier. They also tend to get pregnant more often and are more likely to keep the baby (this was taken from a UK website). What will your son do if she gets pregnant? (Currently, only about two percent of premarital births are relinquished for adoption.) Teenage fathers do not involve themselves very deeply in their new role. Furthermore, the young man often leaves the child’s mother during pregnancy or during the two years after birth. How will keeping the baby affect her future? His? Also, her future ability to educate herself and find a good job are affected by having a baby. Is this what is best for her?
If she gets pregnant, who will pay for the abortion (if she chooses that route) or child care if she keeps it?
If they do have sex, because of her age, there should never be alcohol involved, or her parents might be able to sue regardless of what she says happened.
Should I let her come round to our house again?
Sure. She's an important part of his life. That should make her an important part of yours. In fact, you might even look upon her as a daughter. Which means, if she were your 13 year old daughter, would you want her to be having sex?
Would I let them close the door? No, I wouldn't. Would I supply condoms? No, but I'd make sure he bought them with his own money; if he can't afford condoms, he can't afford any of the responsibilities of sex.