Anatomical tract tracing methods are the gold standard for estimating the weight of axonal connectivity between a pair of pre-defined brain regions. Large studies, comprising hundreds of experiments, have become feasible by automated methods. However, this comes at the cost of positive-mean noise making it difficult to detect weak connections, which are of particular interest as recent high resolution tract-tracing studies of the macaque have identified many more weak connections, adding up to greater connection density of cortical networks, than previously recognized. We propose a statistical framework that estimates connectivity weights and credibility intervals from multiple tract-tracing experiments. We model the observed signal as a log-normal distribution generated by a combination of tracer fluorescence and positive-mean noise, also accounting for injections into multiple regions. Using anterograde viral tract-tracing data provided by the Allen Institute for Brain Sciences, we estimate the connection density of the mouse intra-hemispheric cortical network to be 73% (95% credibility interval (CI): 71%, 75%), higher than previous estimates (40%). Inter-hemispheric density was estimated to be 59% (95% CI: 54%, 62%). The weakest estimable connections (about 6 orders of magnitude weaker than the strongest connections) are likely to represent only one or a few axons. These extremely weak connections are topologically more random and longer distance than the strongest connections, which are topologically more clustered and shorter distance (spatially clustered). Weak links do not substantially contribute to the global topology of a weighted brain graph, but incrementally increased topological integration of a binary graph. The topology of weak anatomical connections in the mouse brain, rigorously estimable down to the biological limit of a single axon between cortical areas in these data, suggests that they might confer functional advantages for integrative information processing and/or they might represent a stochastic factor in the development of the mouse connectome.