We have demonstrated storage of plasmas of the charged constituents of the antihydrogen atom, antiprotons and positrons, in a Penning trap surrounded by a minimum-B magnetic trap designed for holding neutral antiatoms. The neutral trap comprises a superconducting octupole and two superconducting, solenoidal mirror coils. We have measured the storage lifetimes of antiproton and positron plasmas in the combined Penning-neutral trap, and compared these to lifetimes without the neutral trap fields. The magnetic well depth was 0.6 T, deep enough to trap ground state antihydrogen atoms of up to about 0.4 K in temperature. We have demonstrated that both particle species can be stored for times long enough to permit antihydrogen production and trapping studies.