2,151. A Convert is a Full-Fledged Jew

Hilchos Issurei Biah 12:16

If the children who got mixed up were girls, they are both possibly servant girls. Therefore, if a man has relations with either of them, the offspring is a servant out of doubt. Similarly, if a non-Jewish child gets mixed up with a Jewish child, they are both immersed as converts and are considered converts out of doubt.

Hilchos Issurei Biah 12:17

Whenever a non-Jew converts and accepts all the mitzvos in the Torah, or whenever a servant is freed, they are considered Jews in all ways, as per Numbers 15:15, “There will be one law for the community….” A convert may marry within the general Jewish marriage pool right away. Specifically, a male convert or freed servant man may marry a woman who was born Jewish and a Yisroel (i.e., an Israelite) may marry a female convert or a freed servant woman. Converts from four nations are an exception to this rule, namely those from Amon, Moav, Egypt and Edom. If a person from one of these nations converts, he is like a born Jew in all ways except for entering the general marriage pool.