Presta Valve Pin + Core Problem My rear tire Presta valve began giving me some problems this morning. I noticed that the top screw (typically unscrew/screw this to inflate/deflate) was screwed too tightly to the valve pin thread and subsequently to the core. With a pair of pliers I was able to get the screw off but with it came the thread from inside the valve. I know how to screw the core back into the tube thread, but can't figure how to get the valve thread to attach/stay inside the core so that I can put it all back together and inflate the tire?

Have attached a picture showing the tire stem, the vale core, and pin thread/screw so you can understand how things have come apart. 

Any thoughts on how to get the thread back together with the valve core?
The brass rod (I believe its technical name is the "plunger rod") is broken. You can probably find the lower part hanging out in the tube somewhere. Getting it out may be tricky, but you should probably do so before replacing the core.