Results are presented of a new VLA-ROSAT study that probes the magnetic field strength and distribution over a sample of 16 "normal" low-redshift (z ≤ 0.1) galaxy clusters. The clusters span 2 orders of magnitude in X-ray luminosity and were selected to be free of (unusual) strong radio cluster halos and widespread cooling flows. Consistent with these criteria, most clusters show a relaxed X-ray morphology and little or no evidence of recent merger activity. Analysis of the rotation measure (RM) data shows cluster-generated Faraday RM excess out to ~0.5 h Mpc from cluster centers. The results, combined with RM imaging of cluster-embedded sources and ROSAT X-ray profiles, indicate that the hot intergalactic gas within these "normal" clusters is permeated with a high filling factor by magnetic fields at levels of _icm = 5-10(/10 kpc)^-1/2 h μG, where is the field correlation length. These results lead to a global estimate of the total magnetic energy in clusters and give new insight into the ultimate energy origin, which is likely gravitational. These results also shed some light on the cluster evolutionary conditions that existed at the onset of cooling flows.