Can I 1031 Exchange my PR for a portfolio of rentals?

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I'm curious to know if I can do a 1031 exchange with my primary residence to purchase a portfolio of rental properties that are already cash flowing.  My primary residence is worth about $265,000 based on an (exterior only) appraisal I had done recently.  Two comparable homes in my neighborhood sold for about $280,000 however, so that could push mine up to that bracket.  I do have a 30 year fixed conventional mortgage on my house, and have approximately $193,000 left to pay on it.  I don't know if my mortgage will have any effect on a 1031 exchange or not.

My question revolves around how I can do the 1031 exchange.  I understand how it works - I basically trade my property for the value of another at equal or greater value.  But instead of selling my residence and getting just the cash out that I put in originally as the down payment, could I trade the full value of the property ($265k) and use it as a downpayment for a portfolio of cash flowing rentals that I otherwise wouldn't have the cash for?

If there is an REI expert or real estate attorney on BP that can not only answer this, but help me take the next steps to performing this type of transaction, that would be immensely helpful. I've already been getting leads for small portfolios in the range of 5 to 20 houses so I'm very interested if this will work.

Many thanks to whomever can answer this for me!