For some species, like D. melanogaster, there are look-up tables between full-length gene/protein name synonyms and their gene symbols that you could try to parse directly.
More generally, I think your problem is the same as the gene/protein name normalization (GNN) problem, which is currently a matter of active research in the text mining community. If so, then it appears there is no current solution to resolve full length gene/protein names to database identifiers and thence to official gene IDs, as in your case.
The state of the art methods in gene/protein name normalization problem are GNAT and geneTUKit, but they still may not do as well as you like. I also suspect that the same problems experienced by GNN will be experienced by the solutions proposed by Pierre and Michael (which I think are nevertheless both valid and worth trying). For example, running Michael's STRING approach yields the following promising, but not bullet-proof, results:
num_hits official_name_found full_gene_name
1 Dhodh found Dihydroorotate dehydrogenase, mitochondrial precursor
193 Rnf19a not found * ubiquitin-protein ligase
3 Ap2m1 found AP-2 complex subunit mu-1-A
1 MRPL4 not found Proliferation-associated protein 2G4
* several other RNF family members found
Thus you may not get a single hit or the desired gene name with this approach or any other. Unfortunately, inherent variability in gene/protein name usage may be the enemy in the search for a fully automated solution to this problem.
I think a better title for this question might be "Retrieving official gene/protein symbols from full length gene/protein names automatically"
@Casey: Done :)
Thank you all for your comments and suggestions! Having the latest hot computer --> 2500$; One full run of 454 sequencing --> 6000$; Biostar Forum --> Priceless ;)