Could a biological process organically build solid metal "organs"?
That is an excellent question. I was a little confused at first because, without realizing it, you actually cited a perfect example of a biological process that manipulates metal: bone formation.

Bone is made of hydroxyapatite crystal, which is a mineral whose main ingredient is, of course, calcium (metal).

Now, to answer the how of your question; bone is formed, as you eluded, through protein to protein interaction mediated by cells called osteoblasts (osteo- = bone, -blast = immature cell). The osteoblast first locates a site on the bone that is to be expanded, then encases itself in a framework of collagen fibers. When this is done, calcium phosphate and hydroxide bind to the collagen and "harden" as hydroxyapatite (Ca10(PO4)6(OH)2). The osteoblast, now encased in a shell of hydroxyapatite (called the lacuna), transforms into an osteocyte (-cyte = mature cell) and projects long arm-like structures called filopodia which it uses as sensors and to communicate with surrounding osteocytes. All that adds up to look an awful lot like this (with respect to the fact that they are usually fully enclosed, but that would make for quite an insipid picture):



Finally, to put Filipe Rocha's concerns to rest, how is this fully organic and (almost) solid metal structure adapted and "unmade," should the need arise? Well, there is a third type of bone cell called an osteoclast (-clast = to break). Osteoclasts break down bone by secreting hydrogen ions to dissolve the calcium in hydroxyapatite, then cathepsin and other protease to digest the collagen network.

So there you go; a fully biological system with the capability to not only create a non-toxic, solid metal structure, but also maintain it and even manipulate and "unmake" it. Even though bone is not an organ, I'm sure this system could be modified for such an implement if we put our minds to it... :)

Edit:
If calcium isn't "metally" enough for you, you might want to check out limpet teeth, which are comprised of goethite, iron oxide-hydroxide (FeO(OH)). I can go further in depth with the biological processes necessary to produce this mineral if you want; just ask. :)