We have investigated a sample of 28 well-known spectroscopically-identified magnetic Ap/Bp stars, with weak, poorly-determined or previously undetected magnetic fields, with the aim of exploring the weak part of the magnetic field distribution of Ap/Bp stars. Using the MuSiCoS and NARVAL spectropolarimeters we have obtained 282 LSD Stokes V signatures of our 28 sample stars. All stars were detected, showing clearly that when observed with sufficient precision, all firmly classified Ap/Bp stars show detectable surface magnetic fields. To better characterise the surface magnetic field intensities and geometries of the sample, we have inferred the dipolar field intensity and the magnetic obliquity. The distribution of derived dipole strengths for these stars exhibits a plateau at about 1 kG, falling off to larger and smaller field strengths. Remarkably, in this sample of stars selected for their presumably weak magnetic fields, we find only 2 stars for which the derived dipole strength is weaker than 300 G. We interpret this magnetic threshold as a critical value necessary for the stability of large-scale magnetic fields, and develop a simple quantitative model that is able to approximately reproduce the observed threshold characteristics. This scenario leads to a natural explanation of the small fraction of intermediate-mass magnetic stars. It may also explain the near-absence of magnetic fields in more massive B and O-type stars.
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