The micro-image-scanner of the Gregory Coude Telescope at the Observatorio del Teide/Tenerife is used for observations of structures with weak magnetic flux in the solar atmosphere. The methods for the data reduction are chosen carefully to achieve the highest possible performance, also with respect to the needs of handling large amounts of data. A lower detection limit of magnetic flux on the Sun of 2 10(8) Wb on an area of 0 ".67 x 0 ".67 was achieved. Magnetic structures of weak magnetic flux are observed using different spectral lines in network and intra-network regions at disk center of the Sun. The present study deals mainly with a high spatial resolution observation in the magnetically sensitive Fe I 630.25 nm line. We find that weak flux may appear almost everywhere in the granular pattern, with a 65 % preference for intergranular spaces. Some elements have diameters of 2 "-3 " with substructure and distorted shape. Many elements are very small, <1 ". From the ubiquitous appearance of weak magnetic flux structures in the granular pattern, in regions of both upward and downward flow, and in regions of line weakening and line enhancement, we argue that their magnetic pressure may not exceed the granular flow pressure. This gives an upper limit of the amplitude of the magnetic flux density of \ B \ = 5.5 10(-2) T in agreement with the result by Keller et al. (1994)(1,2).