The Torah, itself, does not mention the exact date that the Torah is given, but this can be easily calculated by looking at the story at the beginning of Shemot ch. 20. Summary:
- 1st day of 3rd month (Sivan) they arrive at Mt. Sinai
- 2nd Sivan, Moshe goes up to mountain & G-d tells him that Israel should be a nation of priests, etc. Moses relays this to the people, and they answer, Everything G-d says we will do
- 3rd Sivan, Moshe relays the info to G-d, and G-d says that they should prepare themselves today and tomorrow (i.e. - 3rd & 4th Sivan) and be ready for the 3rd day (i.e. - 5th Sivan.)
- On 4th Sivan, Moshe returns to the people, and decides on his own to add an extra day as a defense in case men had a seminal emission. (i.e., 3 days of preparation, not 2 as G-d originally told him. See Avot D'rAv Nattan ch. 2
Rashi on Exodus 19:3:1 explains that each time Moshe went up the mountain it was on a different day. (I was wondering about that, as from the text, it seems that Moshe might have made more than one trip per day):
"And Moshe went up" - On the second day. And all of his goings up were at daybreak as it is written (Exodus 34: 4) "Moshe arose in the morning."
Thus, that makes the "3rd day" the 6th day of Sivan.
If you match this up with the commandment to count 50 days from the "morrow of the Sabbath", translated as staring from the 2nd day of Pesach, based on the current calculation of the fixed calendar that we now use, that matches with the 6th of Sivan as well.
In the last sentence of @Yishai's amswer, he seems to refer to a time prior to the fixed calendar when each year, the calendar date of Shavu'ot may be different than that of the previous year, depending on the length of Nissan and Iyar. (They could be both 29, 30, or one 29 and the other 30 days long.)