A study is made of nonlinear dust-acoustic waves in a dusty plasma, which consists of any number of cold negatively charged dust grain species, in addition to the presence of the more usual isothermal hot and cold electrons and isothermal positive ions. The Sagdeev potential is obtained in general, together with limits on rarefactive solutions for the dust-acoustic solitons. Weak dust-acoustic solitons can be described by a modified Korteweg-de Vries equation. An application is given to a dusty plasma with one kind of negative grains, in the presence of protons, all electrons having been accreted unto the dust grains. Such dusty plasmas can support rarefactive supersonic solitons, in contrast to the usual ion-acoustic solitons which are compressive. Compressive weak dust-acoustic solitons are less likely to occur. When some streaming is included, one finds a slight decrease in the amplitude of the solitons.