Short answer: One year and 10 days.
Bible Reference: Genesis 7 & 8
The Bible says that Noah was 600 years old when the flood waters came upon the earth. God commanded Noah to bring his family and all the animals aboard the Ark and seven days later, it began to rain and God shut the door to the Ark. The Bible says that it all began “In the six hundredth year of Noah’s life, in the second month, the seventeenth day of the month, the same day were all the fountains of the great deep broken up, and the windows of heaven were opened.” [Genesis 7:11]
It then began to rain for 40 days and nights. As the waters rose, the Ark was borne upon the waters with it. The waters covered all of the mountains by a depth of 15 cubits [about 22 feet 6 inches; the Ark itself was 30 cubits high, or 45 feet tall.]
After it rained that 40 days and nights, the waters prevailed on the earth for another 150 days [5 months] and everything that wasn’t aboard the Ark died. God made a might wind to pass over the earth and the waters began to recede. In fact, the Bible says that at the end of the 150 days of nothing but water, the Ark grounded itself on Mount Ararat. It was the 17th day of the 7th month of that year.
The waters continued to recede for a little over 2 months more, until on the 1st day of the 10th month, the tops of the mountains were exposed.
40 days after the tops of the mountains were seen, Noah sent forth a raven and a dove to see if the flood waters were abated. The dove returned to him, unable to find land, so he waited a week and sent it out again. This time, it came back with an olive leaf in its mouth. He waited yet one more week and sent it out one last time, but it did not return.
In the 601st year, 1st month and 1st day [on Noah's birthday in other words], he looked and saw that the “face of the ground” was dry. But apparently not yet firm. On the 2nd month and 27th day, the ground was dry and God commanded Noah and those aboard the Ark to leave it.
So if we calculate the time that passed between the point where God shut the door of the Ark until God commanded them to leave it:
Year 601 Month 2 Day 27 [End date]
- Year 600 Month 2 Day 17 [Begin Date]
= 1 Year 0 Months 10 Days