I don't know this for sure, but Wikipedia says:

"The exact date of Smith's birth is unknown; however, his baptism was 
recorded on 16 June 1723 at Kirkcaldy."

The source for this is the Scottish novelist and journalist who wrote:

Buchan, James (2006), The Authentic Adam Smith: His Life and Ideas 
<http://www.amazon.com/dp/0393061213> , W. W. Norton & Company 
<http://en.wikipedia.org/wiki/W._W._Norton_&_Company> , ISBN 
0393061213 <http://en.wikipedia.org/wiki/Special:BookSources/0393061213>

Sumitra Shah