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no, at least not a traditional ship. mobile mining outpost, perhaps. but dead mass is a killjoy in space. you'd need big reactor and a lot of propellant mass just to move it from asteroid to asteroid. its probably the way to to mine asteroids actually. leave all the slag in place and move the processed materials. some materials may need to be processed under gravity, unless you can come up with a zero g bessemer process or have a rotating section to provide gravity. you need a lot of heavy hardware to do this though, and a lot of power. definately need megawatt range mpd thrusters to move it around. a reactor that can power those engines would also be able to power the forge while the ship is stationary. its cost to move unrefined ore to the refinery base vs the cost to move the refinery to the ore and the materials back. another approach might be to bring the asteroids to the refinery. set up base on ceres, drop the asteroids onto a pre-designated spot on the surface, and once cool, stripmine the area till its depleted, then go fetch another asteroid, wash rinse repeat.
I still like Nukes idea of throwing a couple of Ion jets onto Apothis as it goes by, coaxing it into orbit and using it as a geostationary locking point for an elevator 
its really the most viable meathod of getting a counterweight. some requirements need to be met however.
1. the object cannot be a rubble pile, a solid metal core asteroid is needed.
2. it needs to cross close to geosync, to save the fuel cost of orbital re-positioning.
3. it needs to be big enough to hold up the cable, but light enough to not snap the cable or do any major damage should it re enter.
it does not do anything for actually making or deploying the cable. lower it from the counter weight, and youd need to place temporary thrusters at various points in the cable to keep it under control as you thread it down to the anchor point. once attached and the cable tensioned (by slightly rasing the asteroid to provide constant centrifugal force on the tether) you could start detaching the control thrusters and attatch your elevator. you might actually favor a multi-tether system, where multiple tethers string to a single object to provide multiple travel paths, you could have low speed cargo haulers and high speed personel transfer lines in both directions to speed up transport.
of course the fact remains, you are gonna need to build it in space. to do that you will need to already have some industrial infrastructure up in space. it would be a money saving endeavor and will only be viable if it can compete with launch vehicles, which will be much cheaper in the future.
