Hi Dan - is short "no" serial passaging is unlikely to generate an immortalised cell from a primary cell. More likely the telomeres of the cells will erode in culture and the cells will become senescent.
One (fairly simple) way to generate an immortal lines is to transform the cells with a plasmid constitutively expressing telomerase (ad select for sub-clones with enhanced growth potential).
From what I've seen telomerase over-expression tend to maintain the phenotype of the primary cell more than the more standard techniques (i.e. SV40 middle T antigen etc).
However I have heard reports that even Tel+ cells are not truly immortal (just have extended growth in culture) and will cease dividing in time.
You could also try over-expressing c-myc and see if this works (although they are more likely to apoptose than grow). I
As a long shot you could try dosing them with sub-lethal bursts of radiation (i.e. UV) but once again probably 99.9999% will just die.