Matt Raymond, Communication Director of The National Archives, talked with CSPAN about Twitter's donation of its entire digital archive of public tweets to the Library of Congress. Highlights of the donated material include the first-ever tweet from Twitter co-founder Jack Dorsey and President Obama's tweet after he won the Presidency.
The LOC is the appropriate place for this, although the National Archives will probably also house all "official" government tweets. As David Ferriero, Archivist of the United States, said in his blog posting:
You might wonder why the National Archives did not acquire the tweets. Our primary purpose is to acquire, preserve, and make available for research the most valuable records of the Federal Government. Because tweets aren’t government records (although tweets of federal agencies can be), the Twitter archive is much better served by the Library of Congress as a cultural institution. At the National Archives, we are working with over 250 Federal agencies and their components to identify and schedule Federal records, some of these most certainly are tweets. Our records appraisal process identifies those records that are valuable enough to be permanently preserved.