Many cold positrons in ultrahigh vacuum are required to produce cold antihydrogen, to cool highly stripped ions, and to study ultracold plasmas. Up to 3.5 × 104 such positrons have now been accumulated into the ultrahigh vacuum of a 4.2 K Penning trap, at a rate exceeding 103/hr. Both the accumulation rate (per high energy positron incident at the trap), and the number accumulated, are much larger than ever before realized at low temperatures in high vacuum. Cooling of high energy positrons (from 22Na decay) in a tungsten crystal near the trap, together with purely electronic trapping and damping, are key to the efficient accumulation and to projected improvements.