A little googling led to this forum discussion, which linked to this forum post and this French-language blog posting. The steps are quite easy, but it's bothersome that Dropbox hasn't made this obvious. The download and installation process is seamless for both other platforms.
- Stop the Dropbox client to your home folder.
- Download the appropriate 1.0.10 linux client for x86 or x86_64.
- Rename or delete the '.dropbox-dist' folder (in a terminal at ~, mv .dropbox-dist .dropbox-dist.backup).
- Unpack the downloaded client archive (tar zxvf dropbox-lnx.x86-1.0.10.tar.gz).
- Restart the Dropbox client from the Ubuntu menu (Applications -> Internet -> Dropbox).
Update 1/31/2011: I installed Dropbox on my new laptop Ubuntu install using the 0.6.7 installer downloaded from the website. After installing, it immediately downloaded an update to the 1.0.20 version. Thanks, Dropbox team.