  We investigate theoretically the phase diagram of a classical Heisenberg
antiferromagnet on the pyrochlore lattice perturbed by a weak second-neighbor
interaction J_2. The huge ground state degeneracy of the nearest-neighbor
Heisenberg spins is lifted by J_2 and a magnetically ordered ground state sets
in upon approaching zero temperature. We have found a new, partially ordered
phase with collinear spins at finite temperatures for a ferromagnetic J_2. In
addition to a large nematic order parameter, this intermediate phase also
exhibits a layered structure and a bond order that breaks the sublattice
symmetry. Thermodynamic phase boundaries separating it from the fully
disordered and magnetically ordered states scale as 1.87 J_2 S^2 and 0.26 J_2
S^2 in the limit of small J_2. The phase transitions are discontinuous. We
analytically examine the local stability of the collinear state and obtain a
boundary T ~ J_2^2/J_1 in agreement with Monte Carlo simulations.
