Your existing lender has to agree to do a short-refi, but if they do then it is certainly possible with a new FHA mortgage. Requirements are explained in detail at http://www.hud.gov/offices/adm/hudclips/letters/mortgagee/files/10-23ml.pdf however the bullet points are:
1. Your new loan amount needs to be within FHA loan limits - which for Saint Joseph County Indiana is $271,050.
2. The homeowner must be current on the existing mortgage to be refinanced
3. The existing first lien holder must write off at least 10 percent of the unpaid principal balance
4. The refinanced FHA-insured first mortgage must have a loan-to-value ratio of no more than 97.75 percent (so if your home appraises for $250,000 then you could get a new FHA loan amount of $244,375)
One item to note, is that the program allows for a 2nd mortgage up to 115% of the home's value (called combined loan-to-value, or CLTV for short). So the current mortgage lender can convert the portion of your loan over the amount needed for FHA financing into a 2nd mortgage, and it's permissible per the loan program.
Not a lot of lenders offer it though, so it is a tough find.
FHA's regular refinance program will do the same - it will have all of the 4 requirements above except for #3, and you cannot put a 2nd mortgage on the home over 97.75% doing what you'd be doing. See http://www.hud.gov/offices/adm/hudclips/letters/mortgagee/files/10-24ml.pdf for those requirements on maximum CLTV when you are refinancing with FHA. More lenders are doing that, including us.
Scott's advice is spot on when trying to get your existing lender to accept a short-payoff. It is very similar to doing a short-sale, so a review of your current situation as well as the reason you are looking/need to do a short refinance will be considered.