Are B Cells unique to an individual's immune system?
Remember first that the human body nearly always, although there are exceptions, rejects any non-genetically identical graft placed into it, and therefore one cannot prepare a general colony of B-cells to confer immunity to the population at large. However, in principle, there is no impediment to removing B-cell progenitors in an individual, conditioning them to produce antibodies (which would be an artificial analogue of the natural process of 'Clonal Selection'), and re-injecting these differentiated B-cells into the patient. To my knowledge this process is currently beyond the capacity of our current cell culture techniques, as B-cell differentiation is controlled by an elaborate cell-signaling process, and even if it was feasible, it would remain impractical for several reasons. First, since the B-cell products, antibodies, can be transferred humorally, and should be tolerated by the general population, many existing treatments simply involve injecting antibodies into the patient's bloodstream. Examples of antibodies taken from volunteers' sera are the RhoGam preparations as well as IgG mixtures used to treat certain forms of immunodeficiency. There is a means of growing B-cells in culture to produce antibodies of precise specificity (i.e. monoclonal antibodies - note that clonal selection occurs in a living host for this technique, the B-cells are subsequently harvested then immortalized in culture). Adalimumab is a therapeutic monoclonal antibody.

There is however, very promising work being conducted involving T-cell vaccination, which is technically similar to what you described. The advantage in using T-cells is that these leukocytes can orchestrate an immune response, mobilizing the several, diverse elements of the immune system to respond to a specific antigen. These techniques are being principally researched as cancer treatment regimens.