The combination of the German Vacuum Tower Telescope and the prototype of ZIMPOL I (Zuerich Imaging Stokes Polarimeter I), a novel, very sensitive imaging polarimeter, has resulted in the first spectra of solar intra-network (IN) fields in circular polarization. The sensitivity in terms of flux density is 0.7 Mx/cm^2. While magnetic fields in solar spots, pores, plages, and in the network have predominantly kG field strengths, the magnetic field strength of the IN flux is a controversial subject due to the absence of direct measurements. We first summarize the current ideas on IN fields and examine previous arguments for their field strength. Our measurements of the magnetic line ratio formed between the amplitudes of the Stokes V profiles of Fe I 5247.1 A and Fe I 5250.2 A are consistent with a field strength well below 1 kG. Since the sensitivity of the magnetic line ratio becomes low for small field strengths, we can only set an upper limit on the field strength of IN fields of 500 G at the level of line formation with a probability of 68% and an upper limit of 1 kG with a probability of 95%. We emphasize that these are the first observations of a magnetic line ratio of unity near disk center.