When you deleted v1 and renamed v2, did you do 'rm -r ProjectName' and 'mv ProjectName2.0 ProjectName' or did you do 'git rm -r ...' and 'git mv ...'?
It sounds like the former, in which case you need to update git to tell it that you've made all the changes. One way to do this would be to do 'git add ProjectName', followed by 'git add -u'. This will ensure that the index is updated such that 1: all added or deleted files in ProjectName are recognised, and 2: all modified or deleted files in the entire repository are recognised.
Alternatively, 'git add -A' will make sure that the index reflects the current state of the working directory, unless you have any suggestions in your .gitignore file.
Having updated the index, you can commit as normal.
If you are having trouble understanding the concept of the index in git, and the way in which file addition/removal is different to simpler VCSs which don't have that concept, you might try reading http://git.or.cz/gitwiki/WhatIsTheIndex or, if you have more time, http://tom.preston-werner.com/2009/05/19/the-git-parable.html. The latter refers to the index as the 'staging area', which is a colloquial term that's becoming quite common as many people find it a more descriptive name.