R' Hirsch understands the אות to be referring back to the previous point - Moshe said (vs. 11) "Who am I that I should ... bring the Children of Israel out from Egypt?" Hashem answered, This that you have asked "Who am I..." is exactly the qualification to be the best messenger. You sense that you lack any qualification for this mission, and that itself will be the sign that I have sent you - it will be clear that the accomplishment was only through Me. Without this, Israel's departure from Egypt would be no different than any other dynamic leader arising to save his people.
The result will be the ability to fully serve Hashem on this mountain.
So the verse should be read and punctuated as follows:
"And He said: [You will be able to take them out] because I will be with you! And the sign that it is because I will be with you is this [your humility]. If you take them out, then you will serve Hashem on this mountain."
The Ohr HaChaim has yet another way to read the verse. Moshe said that he was unfit to stand before Pharaoh and to take the Jews out. Hashem told him that he was fit to stand before Pharaoh, because "I will be with you," and therefore your greatness is greater than his. And the sign that you are fit to take the Jews out is that I am sending you. I wouldn't be sending you if you weren't capable.
The verse should then be read "I will be with you. And the sign [that you are fit to fulfill this mission] is that I am sending you!"
The Rambam (Hilchos Yesodei Hatorah 8:2 (8:4 in Mechon Mamre's count)) understands that the אות is that which the entire nation witnessed Hashem speaking directly to Moshe at Har Sinai:
כך משה רבינו כל ישראל עדים לו אחר מעמד הר סיני ואינו צריך לעשות להם אות. וזהו שאמר לו הקב"ה בתחילת נבואתו בעת שנתן לו האותות לעשותן במצרים ואמר לו ושמעו לקולך. ידע משה רבינו שהמאמין על פי האותות יש בלבבו דופי ומהרהר ומחשב והיה נשמט מלילך ואמר והן לא יאמינו לי. עד שהודיעו הקב"ה שאלו האותות אינן אלא עד שיצאו ממצרים ואחר שיצאו ויעמדו על ההר הזה יסתלק הרהור שמהרהרין אחריך שאני נותן לך כאן אות שידעו שאני שלחתיך באמת מתחילה ולא ישאר בלבם הרהור. והוא שהכתוב אומר וזה לך האות כי אנכי שלחתיך בהוציאך את העם ממצרים תעבדון את האלהים על ההר הזה.
(partial translation) All of B'nei Yisroel were witnesses to Moshe's appointment as messenger... Moshe knew that those who believe based on wonders have incomplete faith and remnants of doubt. Therefore Moshe recused himself by saying they wouldn't believe him ... Hashem told him that after they left and stood on this mountain, all their doubts would be removed... and this is [the meaning of] that which is written "and this is the sign that I have sent you - when you take the nation out of Egypt, they will serve Me on this mountain."
The proof means The proof (with a capital t) - the ultimate proof that Moshe was sent by Hashem will be the first-hand witnessing of Hashem speaking to Moshe, which will happen at this mountain.