Office 365 changing a Distribution List into a shared mailbox?

I am a recent convert to the Office 365 administration tools, and have a bit of a quandry, it may be fairly simple, but then, so am I.

Our Office365 is synched with AD (server 2008), and we have a fairly unruly distribution list for constantly incomming client emails.  the question was posed if we can, through the Office 365 portal, assign the distribution list (which appears in the user and group list) a 365 license to create an actual mailbox, then use the standard powershell command to change it into a shared mailbox.

The parameters are that we want to keep the same email address, and not lose any emails, and the transition would need to take place in roughly an hour after the process is started.

Also, if this is possible, would the members of the distribution list carry over to the shared mailbox?

Thanks in advance for your consideration.