A new way to produce a muon beam for the MuCool Test Area is described. The key idea is to use existing Linac beam-splitting capabilities, based on a fast electrostatic chopper, to split individual 15-Hz Linac beam pulses between the Booster and the MTA, thereby taking advantage of the ability of the Linac to deliver longer beam pulses than the Booster can accelerate. The momentum-analyzing bend at the end of the Linac would be extended so that the 400-MeV H- beam is steered past the momentum dump toward the MTA. Then it would strike a pion production target to generate muons. Such a beam line, located in existing enclosures, could parasitically deliver at least 50 s of spill containing up to 1.4 x 1013 protons to the production target at a 15 Hz rate (i.e. 0.75 ms of spill and 2.1 x 1014 protons per second). Linac pulse length extensions could provide further improvements