The backstory is that the King of the Khazars decided to inquire of the three great faiths in his area: Christianity, Islam, and Judaism. The story as Rabbi HaLevi writes is it a back-and-forth, question and answer approach. It's actually great Shabbat reading because it's only 100 pages.
I'm only halfway through at this point (started reading it late in the day), but I came across a paragraph that is very interesting for a convert. Here is the text as the translation I'm reading says (complete with the poor punctuation):
"If a convert agrees to these rules, he and his children will delight in closeness to G-d. Nevertheless, a convert is not on par with a Jew from birth, because only Jews from birth can attain prophecy. Converts can become sages and saintly men [like Shemayah and Avtalyon, heads of the Sanhedrin and teachers of Hillel,] who were descendants of converts but they cannot become prophets."Aside from the terrible unclear translation/punctuation at the end, have you heard this idea before? What would justify that distinction? And is it related to female converts being ineligible to marry kohanim?
A notable exception to this "rule" (as far as I know!) is Obadiah (also known as Ovadiah), a convert from Edom, a kingdom in modern-day Jordan.