Is Plasmablast a precursor of Plasma cell?
Antigen activated B-cells enter the germinal centre dark zone to form centroblasts which undergo somatic hypermutation. These then form centrocytes in the light zone. The various possibilities/routes the centrocyte can take are shown in the following figure. If the centrocyte undergoes a class switch recombination, it becomes a plasmablast. 

A plasmablast is  defined as 


  Plasmablast
  
  The B-cell lineage precursor of non-dividing plasma cells, which has the capacity to divide and that has migratory potential.


Once it matures as a plasma cell, it is out in the blood secreting soluble immunoglobulins or antibodies.

Another important definition to mentioned here would be of antibody secreting cells.


  Antibody-secreting cells
  
  A term that denotes both proliferating plasmablasts and non-proliferating plasma cells. The term is used when both cell types might be present 


So, yes, they indeed are a precursor of plasma cells.

Also, it is necessary to clarify here that although both plasma cells and plasmablasts are antibody secreting cells, plasma cells lack membrane bound antibodies while plasmablasts retain them.



Finally, this is just the germinal centre response that is shown over here. Plasma cells and plasmablasts can also form in an extrafollicular response in extrafollicular sites in the spleen or in the medullary cords of the liver. That's another story altogether!

For a more detailed explanation, have a look at my other answer.

Image and reference: http://www.nature.com/nri/journal/v8/n1/full/nri2217.html