Leaving science aside for one moment, let’s look at your sentence grammatically. The correct sentence should read:
The fossil is half-Neanderthal half-Homo sapiens.
However, I don’t know why you use the common name for one species (Neanderthal), and the scientific name for the other (Homo sapiens). It would be better to be consistent, for example:
The fossil is half-Neanderthal half-human.
The fossil is half-Homo neanderthalensis half-Homo sapiens.
Now, back to science. A fossil cannot be two species at the same time, it is either one or the other (in this case, Neanderthal or human). It cannot be both. Results of DNA investigations into Homo neanderthalensis were recently released, and it was concluded that there was interbreeding between Neanderthals and humans. This has led many experts to state that Neanderthals and humans are both subspecies of Homo Sapiens (i.e. Homo sapiens neanderthalensis and Homo sapiens sapiens), rather than two distinct species as was previously thought.