You're trying to feed the cable through the wrong way.

Most universal brake cables come with two ends. One is a short, round cylinder stuck sideways on the cable. That's a mountain end, and not what you need. The other is kinda bullet shaped, that's the one you want. Snip off the mountain end and feed the cable through the rotating bit on your aero lever, from the front. Carefully feed the end of the cable through the hole in the back of the lever (you may need a flashlight to see it) and it should pop out of the hole on the back of your lever. The bullet shaped end will seat into the larger hole on the rotating bit in your lever. You will have to pull the cable out of the housing to do this.

If you don't have the tools, I would really recommend letting a LBS do it. If they're not busy, see if they'll let you watch. It's very easy for the cable to fray into an unusable mess when cut improperly.

Thanks, I really appreciate the detailed description. I do have the tools required, so I'll try this at home before going through the trouble of taking my brakeless bike to an LBS.

Look again. The picture of the cable apastrak attached is a derailer cable. It's a dual cable for mountain bike/road shifters which use the same end and a 3 speed/old Schwinn shifter which used a flat disc end that looks like a mountain bike brake cable end but is much smaller.

The attachment in a later post, shown below, illustrates the difference.

This is good advice. The cable manufacturers have done you a favor in that they have provided you with two possible ends. You just have to cut off the one you don't need. If you cut the cable with the wrong tool or badly you will end up with a frazzle you can't thread through. On the spot you want to cut the cable you can dab it with super glue or heat it with a small torch and solder the wire together, then cut through the glue or solder.

This is a really easy job when you know how, but it is worth while letting somebody who knows how show you the technique.