Tricky, but do-able with 4 possible options:
Option 1: Is the easiest, but only open if your current hosting provider allows your to edit DNS for your domain. Simply add an "A" type DNS record for "beta" to point to the server IP address that DH assigns to your domain, leaving all of the current DNS info in tact - in about 12-48 hours the site should be running with no interruption in any service.
Option 2: Move everything to DH and use DH's DNS editing capacity to add an "A" type DNS record for "www" to point back to the server IP address assigned to your domain by the "old" hosting provider. You also have to decide which hosting provider you want to handle e-mail and such, but if you leave all that to the DH servers, you'll only need to fiddle with the one DNS record. This option will cause a 12-48 hour disruption of service while the new DNS propagates.
Option 3: GoDaddy offers "Total DNS Control" for domains that are "parked" on their service. You'll have to park the domain at GoDaddy and edit all of your DNS records to point to the appropriate IP addresses for your old hosting provider and DH. This option will also cause a 12-48 hour disruption of service and if you're not familiar with how DNS works, the chances of screwing it up and creating a much longer disruption is fairly high.
Option 4: If the "beta" site is only for you and a limited number of people, you (and they) can edit their local "hosts" file to point "beta.whatever.com" with the server IP address that DH assigns to your domain - you leave all current domain info in tact, there's no disruption of service, and the results are immediate.
So now: Choose wisely, young grasshoppah.