Granted it’s harder to take an existing site and backward engineer it for mobile rather than starting with a clean slate but if you build a separate site for mobile then you have the extra headache of 2 sites and all that content to manage.
I feel it would be better to redesign the current one so that it works across devices from the start and in the process make it a better experience for all devices.
As Bruce Lawson said a while ago the question is "whether there is such a thing as a mobile web or is there only one Web that we access using a variety of different devices?
Where does the mobile web start and where does it end and how do you detect them? We have devices ranging in size from small phones, large phones, tablets of multiple sizes, laptops, desktops etc. Users typically access the web on various devices depending where they are it seems that the consensus of opinion is that visitors should experience more or less the same thing but in a way that suits the platform.
We should build desktop sites that are fast and easy to use from the start and then converting them to mobile (or vice versa) isn’t so much of a problem.
Apps are a separate concern as an app usually does one thing and does that one thing well but does not really relate to the web as a whole.