This is an example of underspecification. English, Hebrew, etc., do not have temporal information associated with nouns. There are languages that have nominal tense, so that something like "ex-president" and "president-elect" (or "ex-virgin" and "future-virgin") could be encoded grammatically, but that's not the case here. So yes you are correct. Otherwise it could also indicate that someone is permanently a virgin, such that Jesus's virgin birth would be nullified if Mary ever had sex after his birth too.
However, as you point out the sentence is ambiguous (actually, it's vague, underspecified, because it isn't strictly a set of particular alternatives, unless you assume a finite set of times at which to evaluate "the virgin"), and it may still be the case that the meaning of "virgin at the time of birth" was intended. This is a matter of interpretation and context, not inherent in the semantics. The literary and cultural context, as well as other verses, might clarify this. For one thing, given the attention this has gotten, wouldn't it be weird to mention she's a virgin before becoming not a virgin inherently by the information in the sentence?
Note also that the sentence could be interpreted as her becoming not a virgin while already pregnant with Jesus who was conceived immaculately.
In other words, this sentence is up to interpretation. The question is how the words were intended to be interpreted, not how they could possibly be interpreted.
I will say that generally the easiest interpretation is one of consistency, which also explains why nominal tense only rarely develops (contrastive use is rare, even though it would be useful in those cases). That is, assuming that "the virgin" applies throughout the moment of speech, the event described, and so forth, makes sense. A "permanent" reading, unless otherwise indicated, is a reasonable default. So personally I'd be more likely to focus on the ambiguity of "virgin" vs. "young woman", because those seem the most likely to be used in these circumstances. (It would make little sense and be of little relevance to say "the current virgin [who will later not be a virgin after conception] will give birth".)