We calculate the rates of elastic and inelastic collisions in a gas of spin-polarized triplet helium ({sup 4}He{sup {asterisk}}{up_arrow}) in a wide range of temperatures (0{lt}{ital T}{lt}0.5 K) and magnetic fields (0{lt}{ital B}{lt}100 kG), which includes all values of {ital T} and {ital B} relevant for trapped {sup 4}He{sup {asterisk}}{up_arrow}. At densities {ital n}{approx_lt}10{sup 13} cm{sup {minus}3}, where three-body recombination is not important, the main inelastic processes are spin relaxation and relaxation-induced ionization, both induced by the spin-dipole interaction in pair collisions. For ultralow temperatures {ital T}{approx_lt}10 mK and low magnetic fields {ital B}{approx_lt}100 G, the leading decay channel is relaxation-induced ionization. If either {ital T} or {ital B} is higher, the gas decays through spin relaxation. The ratio of elastic to inelastic rate is found to be large even in the nanokelvin regime, which is promising for evaporative cooling of {sup 4}He{sup {asterisk}}{up_arrow}. {copyright} {ital 1996 The American Physical Society.}