  We have studied the evolution of the weakly first order antiferromagnetic
transition in heavy fermion Yb$_3$Pt$_4$ using a combination of specific heat,
magnetic susceptibility, and electrical resistivity experiments. We show that
magnetic fields suppress the Neel temperature, as well as the specific heat
jump, the latent heat, and the entropy of the transition, driving a critical
endpoint at 1.2 K and 1.5 T. At higher fields, the antiferromagnetic transition
becomes second order, and this line of transitions in turn terminates at a
quantum critical point at 1.62 T. Both the ordered and high field paramagnetic
states are Fermi liquids at low temperature, although the former has a much
larger magnetic susceptibility and stronger quasiparticle scattering. Unlike
previously studied quantum critical systems, the quasiparticle mass in
Yb$_3$Pt$_4$ does not diverge at the quantum critical point, implying that here
the quasiparticle interactions drive the zero temperature transition. The Fermi
liquid parameters are nearly field-independent in the ordered state, indicating
that the fluctuations never become fully critical, as their divergences are
interrupted by the first order antiferromagnetic transition.
