A primordial cosmological magnetic field induces and supports vorticity or Alfvén waves, which in turn generate cosmic microwave background (CMB) anisotropies. A homogeneous primordial magnetic field with fixed direction induces correlations between the al-1,m and al+1,m multipole coefficients of the CMB temperature anisotropy field. We discuss the constraints that can be placed on the strength of such a primordial magnetic field using CMB anisotropy data from the Wilkinson Microwave Anisotropy Probe experiment. We place 3 σ upper limits on the strength of the magnetic field of B<15 nG for vector perturbation spectral index n=-5 and B<1.7 nG for n=-7.