If you have not had PI there in the past, it is probable a seed that has sprouted and doesn't have an enormous root system yet. Have someone who is not particularly sensitive to PI pull the vine and dispose of it in a plastic bag. Or... Tell a fishing friend that you need a cheap long-handled 'fish-hook disgorger'. wearing plastic gloves, use the disgorger to pull the vine. Drop the vine and tool into a plastic trash bag and dispose of both.

I'm battling PI in my front yard too. It was intertwined at the base of a large butterfly bush, so we cut back the BB Bush in the fall and removed the PI (carefully). Now it's back in the same bed. I took undiluted glycophosphate (generic round up) and soaked a paper towel in it and wrapped it around the vines which were just popping out of the ground. I'm also doing this for some crossvine that I attempted to remove, but it's coming back.

Just be careful. I use disposal plastic gloves and then toss them.

i use one of those disposable "foam" paintbrushes.
put a drop or two of dishwashing liquid (will help it stick to the surface of glossy leaves) in some undiluted herbicide (like roundup or brush-b-gone) and paint some of the leaves with that, being careful to not let it drip onto foliage of nearby plants. you will need to remove the vines after they die. those can still contain the toxic oils so wear gloves while doing that or use a hoe or cultivator to dig them out.

Unfortunately, the vines will have to be handled at some point, even if you kill it first with Jeff's suggestion.

Before you start removing the vines, go to your local drug store or Wal Mart and get some stuff called Technu. It's a product to bind any poison ivy oils that may have gotten on your skin.

When you remove the vines, wear gloves and long sleeves. Long pants too. Tie back your hair to keep it from touching the vines, and also to keep you from brushing it back with contaminated hands while you work. Be careful not to touch your skin at any time with your gloves or any other clothing that has been exposed. As soon as you are done, strip off all your clothes and wash them in hot water. (I usually wash mine a second time, just to be sure.)

Then, immediately use the Technu, and treat any areas you even think might have been exposed. Face, neck, wrists and arms, etc. Heck, treat your whole body just to be sure. Shower afterwards.

I'm very allergic to poison ivy, and have even been to the ER with severe breakouts. But a few years ago someone recommended Technu to me, and I've never had a breakout if I've used Technu immediately after exposure.

In a pinch, if you've been exposed to poison ivy and don't have Technu handy, wash with liquid dish detergent (such as Dawn) and hot water. Don't use regular soap, because it just pushes the oils around on your skin and can actually make things worse. You want a detergent that will cut the oils and wash them away.

One last thing... wash your tools with detergent and hot water, including the handles. The poison ivy oils will remain on them if you don't, and you could end up contaminating yourself again, even months later. Wash your shoes too. And your hat. And anything else that might have come in contact with the vines.

Oops, I misspelled it ... it's Tecnu.