I'm so sorry for your loss. I had a similar experience -- first u/s, no heartbeat. We were (and still are) devastated.
It's likely that your OB will offer you a few different options:
First, you may elect to wait to let your body process the miscarriage naturally. This could take a few days or weeks, and you really have no control over when/where this happens. Many people prefer this option as it is the most natural and usually carries less risk. However, it can be a physically painful process, and because you may have to wait a while, it can take some time to get the closure you may need. And, if your body doesn't pass everything naturally, you may still need one of the other options.
Second, your doctor may offer you medication to expedite the miscarriage. You take this at home, and within a few hours you will start to bleed, cramp, and pass clots and tissue. From what ladies have posted on here, this can be quite painful, but usually is over pretty quickly. You have more control over where/when (you can make sure you are at home and take the medication at a time that is convenient to you), but some people still don't pass everything even with medication.
Third, your doctor may offer you a minor surgical procedure called a D&C or D&E. This is usually done under heavy sedation or general anesthesia, but usually takes only about 20-30 minutes. It carries the risk of any surgery, but for most people is the most physically painless option. Again, you have control over where/when it happens, but obviously you are in a hospital or outpatient facility, and not at home (though most people are discharged within an hour or so of the procedure).
Our doctor was able to get me in for a D&C on the same day as the u/s that confirmed our loss. She told us that this was our best bet of being able to start trying again soonest (most docs say to wait 1-3 cycles -- ours told us 1). I think it was the right decision for us, and I don't regret it.
Whatever options you are given, and whatever you end up choosing, most of us agree that the emotional pain is harder to deal with than the physical pain. Personally, this is one of the most difficult things I've ever been through. Please give yourself time to grieve and heal emotionally. We are all here to help in any way that we can.