Why does a ScheduledExecutorService not run a task again after an exception is thrown?

For executing periodical tasks, I looked at Timer and ScheduledThreadPoolExecutor (with a single thread) and decided to use the latter, because in the reference for Executors.newSingleThreadScheduledExecutor(), it says:

Note however that if this single thread terminates due to a failure during execution prior to shutdown, a new one will take its place if needed to execute subsequent tasks.

My plan was to use this as a safeguard against uncaught exceptions in a watchdog piece of code that I want to monitor other operations. I wanted to make sure and wrote the test below, which promptly failed. It seems I was making wrong assumptions, or is something wrong about my test?

Here's the code:

@Test
public void testTimer() {
    final AtomicInteger cTries = new AtomicInteger(0);
    final AtomicInteger cSuccesses = new AtomicInteger(0);

    TimerTask task = new TimerTask() {
        @Override
        public void run()
        {
            cTries.incrementAndGet();
            if (true) {
                throw new RuntimeException();
            }
            cSuccesses.incrementAndGet();
        }
    };

    /*
    Timer t = new Timer();
    t.scheduleAtFixedRate(task, 0, 500);
     */
    ScheduledExecutorService exe = Executors.newSingleThreadScheduledExecutor();
    exe.scheduleAtFixedRate(task, 0, 500, TimeUnit.MILLISECONDS);
    synchronized (this) {
        try {
            wait(3000);
        } catch (InterruptedException e) {
            e.printStackTrace();  //To change body of catch statement use File | Settings | File Templates.
        }
    }
    exe.shutdown();
    /*
    t.purge();
     */
    Assert.assertEquals(cSuccesses.get(), 0);
    Assert.assertTrue(cTries.get() > 1, String.format("%d is not greater than 1. :(", cTries.get()));
}