How can I slow down a cart without causing it to stop?

Unless your minecart has a lot of momentum (not speed), powered off minetracks will stop your minecart. This wasn't the case.

The solution was placing powered minetracks right before, but I got the quantity wrong. Even with generous amounts of powered minetracks at launch, I needed four or five powered minetracks to barely offset a single powered off. That's perhaps because, since you stay on a powered off track for longer (you're slowing down after all), they cause more of an impact on your speed and momentum.

In my case five powered minetracks did the trick. Now the minecart walks those four pieces of track very slowly, giving passengers a chance to get their shot of nostalgia, and then picks back up the pace with more powered rail.

This approach isn't cheap, obviously, if you're not using /give or /i.

Another solution that's very simple is this:

Having a one block drop with a gap like this will slow your cart right down to a crawl, but it will still continue quite a ways, provided there's a passenger in it.

There is one technique that you can use that is a little glitchy, but it works. If you hook up a segment of powered rail to a very fast clock (made with repeaters so it doesn't burn out), the powered rail will quickly alternate from powered to unpowered, and minecarts passing over it will travel slowly.

Here is what the setup looks like (both repeaters are on the lowest setting):

To start the repeater clock, you can put a Redstone Torch down like so, and then destroy it.

Edit: Notch broke fixed this kind of clock, but there is an alternative.

The block in this image is powered by the Redstone Torch & wire behind it. When the Redstone Torch (highlighted by my cursor) is placed, it will turn off almost instantly (because the block it is placed on is powered), sending a quick pulse to the clock.