The authors have analysed the linear Zeeman effect (LZE) for Rydberg states of the H₂ molecule. Three distinct angular momentum coupling schemes are found to be appropriate for different regions of the molecular wavefunction. Using quantum defect theory the authors have calculated energy level positions and line intensities for n>5 for photoabsorption from the ¹ Sigma _g⁺ ( nu =0, J=0) ground state of H₂. The authors find that: (i) whereas in the absence of a magnetic field the dipole selection rule restricts upper states to be pure J=1, the removal of J conservation by the field means that transitions to states dominated by J=2, 3, 4, 5, . . . also become accessible in the range B=1-30 T; (ii) The M degeneracy is removed by the field. There are wide variations in behaviour between corresponding lines for different M.