I highly do not recommend doing that. As is evidenced with troubles other members have been having, if you register the domain with the host, you'll likely have trouble recovering the domain if there's a problem with the host (such as their sudden disappearance).
If you want to setup a site, for your own sanity and to make your lives easier, register the domain with a reputable registrar (godaddy, network solutions, register.com, etc...) before getting your hosting, don't do it at the same time.
Fair enough.
But, erm, why is a registration company less likely to fall bankrupt, or go out of business, compared to a hosting company, especially one with a record of relative longevity such as DreamHost?