All problems appear to be resolved. Users who downloaded the corrupt update should now be able to download a working update.
UPDATE: 07/06/12 1:30 EST
It appears this has now been resolved and you can now download the update.
Still no information as to what you should do if you already downloaded the corrupt update.
This problem ONLY effects current customers updating to the new version, NOT new downloads.
If you experience problems after installing the update please do not delete the app unless you are okay losing any saved bookmarks or checklists as well as having to re-sync everything that is in your Dropbox. That being said, it does appear that deleting the app then re-installing fixes the problem.
If you are not comfortable with that please stay tuned here.
More info here:
I will update this when I have more information.
As usual, Apple is not saying anything other than that they are working on it.