What happens when a thread with active event handler exits ?

I feel that using the relatively new async feature of C# is the best way to implement asynchronous methods because it provides an exit point for the method when it completes. You can use it like this

//subscribe to the event
button.Click += MyEventHandlerAsync;
// When the event fires this handler is called
private async void MyEventHandlerAsync(object sender, RoutedEventArgs e)
       {
           //the UI  thread starts the task here and then returns
           //the task runs on the threadpool
           string text = await Task.Run(() => this.MyMethodThatReturnsAString());
           //When the task finishes, control returns here
           //Do something with the result
           //Set a label on the UI thread
           this.label.Content = text;
       }

       private string MyMethodThatReturnsAString()
       {
           //Long running method that finds the meaning of life
           //........
           return "Forty Two ";
       }