I first found this account on J. L. Bell's Boston 1775 blog in the post, "Sheriff Greenleaf and Col. Crafts read the Declaration."  So I dug in to see what else I could find.
On 25 July 1776, the Continental Journal and Weekly Advertiser reports:
"Thursday last, pursuant to the orders of the honourable Council, was proclaim'd from the Balcony of the State House in this town, the DECLARATION of the AMERICAN CONGRESS, absolving the United Colonies from their allegiance to the British Crown, and declaring them FREE and INDEPENDENT STATES. ...
At One o'Clock the Declaration was proclaimed by the Sheriff of the County of Suffolk, which was received with great Joy, expressed by three Huzzas from a great Concourse of People assembled ..."
Bell's post mentions and quotes a letter by Daniel Greenleaf (1762-1853), William Greenleaf's son, published in the Boston Transcript on 2 Aug 1855, in which Daniel Greenleaf recollects his father reading the Declaration of Independence. Bell states that Daniel Greenleaf wrote the letter in October of 1841, and given that Daniel died in 1853, the letter would had to have been written prior to its publication in the Boston Transcript. As Daniel Greenleaf would have been several months shy of 14 years of age in July 1776, he might well remember the events of 18 July 1776.
Daniel Greenleaf's letter, as published in the Boston Transcript, states in part:
"The Declaration of Independence was read by William Greenleaf, (my father,) then sheriff. ... My father was so proud of that proclamation that he had the paper from which he read it framed and glassed and it hung over his parlor fireplace as long as he was a housekeeper. As his voice was rather weak, he requested Colonel Crafts to act as his herald; they stood together at the front of the balcony, and my father read a sentence, which was immediately repeated by Crafts, and so continued to the end, when was the huzza, as mentioned." 
 "In Congress, July 4, 1776. A Declaration by the Representatives of the United States of America, In General Congress Assembled." The Continental Journal and Weekly Advertiser, 18 July 1776, pg. 1. Online, in "The Coming of the American Revolution," Massachusetts Historical Society (http://www.masshist.org/revolution/doc-viewer.php?old=1&mode=nav&item_id=113 : accessed 4 July 2012)
 Bell, J. L. "Sheriff Greenleaf and Col. Crafts Read the Declaration," Boston 1775, 4 July 2007. Online (http://boston1775.blogspot.com/2007/07/sheriff-greenleaf-and-col-crafts-read.html : accessed 4 July 2012)
 Continental Journal (Boston, Mass.), 25 July 1776, pg. 2. Online. Early American Newspaper Series 1, 1690-1876 (Newsbank). Accessed 3 July 2012.
 "Reminiscences of an Old Bostonian," Boston Evening Transcript, 2 Aug 1855, pg. 1. Online. Google News (http://news.google.com/newspapers?id=H4Q-AAAAIBAJ&sjid=B1oMAAAAIBAJ&pg=2091%2C649929 : accessed 20 Jun 2012)