The random field Ising model in three dimensions with Gaussian random fields is studied at zero temperature for system sizes up to 603. For each realization of the normalized random fields, the strength of the random field, Δ and a uniform external, H is adjusted to find the finite-size critical point. The finite-size critical point is identified as the point in the H−Δ plane where three degenerate ground states have the largest discontinuities in the magnetization. The discontinuities in the magnetization and bond energy between these ground states are used to calculate the magnetization and specific heat critical exponents and both exponents are found to be near zero.