  The Monte Carlo with Absorbing Markov Chains (MCAMC) method is introduced.
This method is a generalization of the rejection-free method known as the
$n$-fold way. The MCAMC algorithm is applied to the study of the very
low-temperature properties of the lifetime of the metastable state of Ising
ferromagnets. This is done both for square-lattice and cubic-lattice
nearest-neighbor models. Comparison is made with exact low-temperature
predictions, in particular the low-temperature predictions that the metastable
lifetime is discontinuous at particular values of the field. This discontinuity
for the square lattice is not seen in finite-temperatures studies. For the
cubic lattice, it is shown that these `exact predictions' are incorrect near
the fields where there are discontinuities. The low-temperature formula must be
modified and the corrected low-temperature predictions are not discontinuous in
the energy of the nucleating droplet.
