OK, I'll be repeating some of what I said before about budgeting, but now I think I'm on the same page as you.

If you're only going to run a front brake, you can probably get the cable for about $5. You need cable and housing, but if you go to your LBS, they'll be happy to cut you a two-foot piece from their bulk supply and sell you that and an inner cable. Make sure they give you a couple of ferrules and a crimp too.

You can reuse your current brake levers and probably the caliper (if you replace the 27" wheels with 700c wheels, you could have brake reach issues, but hopefully not). So you only need a pair of brake pads. I used linear brake pads to replace the pads on my Dia Compe center pulls, but I'm not sure if that will work with all calipers. You can get cheapo pads for $5 or top-of-line Kool Stops for $12, so I'd spend the extra bucks for the good stuff here.

If you get a 1/8" chain, you avoid some possible incompatibilities with some track cogs. You can get a KMC Z410 for $8 or less. If you happen to have a 6-, 7- or 8-speed chain lying around, you can use that, though it may be 3/32" which raises the cog compatibility question.

Whether or not you can or should re-use the existing cranks is an open question. If you can unbolt at least one of the chainrings (or if you're willing to leave both on there and be laughed at), you can re-use them. You'll almost certainly want to replace them at some point, but I understand you're trying to go on a shoestring budget here. So let's assume you will reuse it.

If you keep your current bottom bracket, you'll probably need to overhaul it. Buy some grease if you don't have it. Everybody here loves Phil Wood, but I use Park PPL-1, which is much cheaper -- a $5 tube will cover the whole project and then some. You'll also need some ball bearings, probably 1/4". This is an easy procedure and you definitely don't want to skip it. (BTW, don't remove the fixed cup on the bottom bracket.) You'll also want to do this with the headset, which will use a different size ball bearings. These things generally need to be done every year or so with loose bearing parts, so with an old bike it's certainly going to be needed. Between grease and ball bearings, these jobs will cost you $10 or less total.

If you keep your current bottom bracket you'll also need to solve chainline issues with the rear wheel.

If the wheels you have are in good shape, you could use them. The main problem is that they're almost certainly steel, and rim brakes don't work with steel wheels in the rain -- not to mention they're insanely heavy. The other problem is that while you can thread a fixed cog onto them, you can't thread a lockring on. This means that you run the risk of inadvertently unscrewing the cog when applying back pressure on the pedals, which you'll probably be doing because you're trying to stop, which means that's a really bad time to have the cog come off. That said, people do this. Since you'll be running a front brake, it's not quite crazy (unless it's raining).

If you buy wheels, they'll almost certainly be setup for a 42 mm chainline, which is not likely what you'll get with your current crankset and bottom bracket. Measure the chainline on the crankset. If it's only off by a few millimeters you could conceivably fix it by putting a spacer behind the cog. More likely it'll be more than that, which puts you back in the respacing+redishing camp. In theory, you can do this yourself. In practice, unless you've built wheels before, I'd take it to the LBS. There's a place around here that'll do it for $35. As you ponder that, consider that you can fix this problem with a new crank and bottom bracket for about $75 in which case you wouldn't have to pay another $35 to have it undone if you ever wanted to upgrade the crank.

Whether you buy wheels or use the current wheels, you'll need a track cog. Consider your gearing ratio. Since your using the existing crank, you'll probably have a 52 or 53 tooth chainring, so you may want a cog as big as 20 teeth. Most wheelsets that come with cogs come with 15 or 16 tooth cogs, but if you talk to the seller, you may be able to get this swapped out at the time of purchase. If you're buying wheels, you'll want a lockring, which may be included. If the wheels don't come with rim tape, you'll need that too, $3-$5.

If you're open to singlespeed, as opposed to fixed, there is another option available to you. You can buy a set of wheels meant for a geared bike (you can get a set of Alex/Dimension wheels for around $75) and buy a single-speed conversion kit. The conversion kit will come with spacers, a lockring (not the same as the one discussed above) and one or more cogs. The one from Performance Bike comes with 16-, 18- and 20-tooth cogs, so that would work for you. The nice thing about this is that you can use the spacers to get your chainline exactly where you want it. The downsides are that you're carrying around a bit of extra weight in the form of the big freehub, you can't go strictly fixed and it doesn't look as clean.

You almost certainly need new tires and tubes. You can get $15 tires each or less, but for $26 each you can get tires with decent flat protection. You can mail order tubes for $3 each, but at your LBS they'll likely be $7.

Somebody here recently posted a deal with wheels, tires, tubes, cog and lockring for $130. The tires weren't great, but it's hard to beat that deal. I recently bought a set of Formula/Mavic CXP22 wheels with

Make sure you can adjust the seat and stem height before you get into any of this. Those things are notorious for seizing on old bikes. If they're good, you can keep the seatpost and saddle if you're happy with them.

Finally, I do want to repeat that handlebar tape is the single cheapest thing you can get to make an old bike feel newer. You can get cheap stuff around $6 or good stuff around $12.

You might be able to get this all built for around $200.

I recently built up an old Gitane as a singlespeed for about $80 using wheels (with spacers as described above), chain, saddle, tires and tubes that I had sitting around my garage, but I've been upgrading it constantly and currently have almost $500 in it. I'm never satisfied.

So, anyway, shopping list...

Brake cable and housing
Brake pads
Chain
Grease
Ball bearings for bottom bracket
Ball bearings for headset
Wheels + Cog + Lockring (or Wheels + conversion kit)
Tires
Tubes
Handlebar tape