  The structure of the B-L MSSM theory--specifically, the relevant mass scales
and soft supersymmetric breaking parameters--is discussed. The space of initial
soft parameters is explored at the high scale using random statistical sampling
subject to a constraint on the range of dimensionful parameters. For every
chosen initial point, the complete set of renormalization group equations is
solved. The low energy results are then constrained to be consistent with
present experimental data. It is shown that a large set of initial conditions
satisfy these constraints and lead to acceptable low energy particle physics.
Each such initial point has explicit predictions, such as the exact physical
sparticle spectrum--which is presented for two such points. There are also
statistical predictions for the masses of the sparticles and the LSP species
which are displayed as histograms. Finally, the fine-tuning of the $\mu$
parameter--which is always equivalent to or smaller than in the MSSM--is
discussed.
