The possibility of pervasive weak selection at tens or hundreds of millions of sites across the genome, suggested by recent studies of silent site DNA sequence variation and divergence, raises the problem of the survival of the population in the face of the large genetic load that may result. Two alternative resolutions of this problem are presented for populations where recombination is sufficiently frequent that different sites under selection evolve independently. One invokes weak stabilizing selection, of the magnitude compatible with abundant silent site variability. This can be shown to produce only a modest genetic load, due to the effectiveness of even weak stabilizing selection in keeping the trait mean close to the optimum. The other invokes soft selection, whereby individuals compete for a limiting resource whose abundance determines the absolute fitness of the population. Weak purifying selection at a large number of sites produces only a small variance in fitness among individuals within the population, due to the fact that most sites are fixed rather than polymorphic. Even when it produces a large genetic load, it is compatible with the observations when selection is soft. It may be very difficult to distinguish between these two possibilities.