Use can also use the old French way of securing the ends of bar tape- very fine hemp twine.
Here is one of my bikes with this treatment:
http://harmonias.com/bikes/Luna_5.jpg
http://harmonias.com/bikes/Luna_6.jpg

One of my husband's bikes with the same twining treatment:
http://harmonias.com/bikes/B_Ramb_4.jpg

Here's a particularly fine example elsewhere:
twined tape ends

There are instructions in a few places on the web as to how to do this.
You can either leave the twine plain afterwards, or paint it with a coat of clear shellac to keep in real tight and neat.