My opinion: there is no deeper meaning in that sentence (or verse) at all. The songwriter just phrased it that way for the rhyme, and because it has an air of a meaning (because of Goethe's Gretchen), but that's just pretending.
Evidence to support this hypothesis is in the other answer here: everybody seems to be struggling to identify a meaning that would go beyond the rather superficial (and obvious) "It seems to refer to Faust's sweatheart". But no concrete further characterisation of Faust's Gretchen seems to fit in here.
Now, as the human mind is predisposed to search for (and construct) meaning in what accidential pattern ever it faces, here is an interpretation of the verse that at least would allow for a meaning:
The verse could be read as:
Die schwarze Wissenschaft ist wie ein reifes stolzes Mädchen:
hast du sie, dann zähme sie; wenn nicht, wird sie zum Grätchen
where Grätchen is a small Gräte (fishbone), i.e. something that seems innocuous but can kill you if not handled with care.
This at least would fit into the idea of the verse that "proud girls" need to be "tamed", i.e. held under control. (I do not subcribe under this message, I am just reporting it.)
Note that I do not claim that my Grätchen interpretation is what the author had in mind. I just suggest it as a way to make sense of an otherweise potentially meaningless (or poor-in-meaning) verse.