Firstly, you could call DB.SubmitChanges() right after stats.Visits++, and that would greatly reduce the problem.
However, that still is not going to save you from the concurrency violation (that is, simultaneously modifying a piece of data by two concurrent processes). To fight that, you may use the standard mechanism of transactions. With LINQ-to-SQL, you use transactions by instantiating a TransactionScope class, thusly:
using( TransactionScope t = new TransactionScope() )
{
var stats = DB.Stats.First();
stats.Visits++;
DB.SubmitChanges();
}
Update: as Aaronaught correctly pointed out, TransactionScope is not going to help here, actually. Sorry. But read on.
Be careful, though, not to make the body of a transaction too long, as it will block other concurrent processes, and thus, significantly reduce your overall performance.
And that brings me to the next point: your very design is probably flawed.
The core principle in dealing with highly shared data is to design your application in such way that the operations on that data are quick, simple, and semantically clear, and they must be performed one after another, not simultaneously.
The one operation that you're describing - counting visits - is pretty clear and simple, so it should be no problem, once you add the transaction. I must add, however, that while this will be clear, type-safe and otherwise "good", the solution with stored procedure is actually a much preferred one. This is actually exactly the way database applications were being designed in ye olden days. Think about it: why would you need to fetch the counter all the way from the database to your application (potentially over the network!) if there is no business logic involved in processing it. The database server may increment it just as well, without even sending anything back to the application.
Now, as for other operations, that are hidden behind // ..., it seems (by your description) that they're somewhat heavy and long. I can't tell for sure, because I don't see what's there, but if that's the case, you probably want to separate them into smaller and quicker ones, or otherwise rethink your design. I really can't tell anything else with this little information.