I am in the process of converting my IRA to a Roth IRA. I am also changing to an on-line broker for the Roth IRA. My current broker has made a mistake, and instead of transfering the funds, they sent me a distribution check. Can I deposit these funds directly into a Roth IRA account, or will I be subject to the 10% penalty because my broker didn't properly convert the funds?

You have 60 days from the date of distribution to roll the funds to the Roth IRA. If rolled within the 60 day period, you won't have to pay the 10% premature distribution penalty.