Diffie-Hellman key exchange is a protocol but does nothing about authentication.
There is a high-level, conceptual way to see that. In the world of computer networks and cryptography, all you can see, really, are zeros and ones sent over some wires. Entities can be distinguished from each other only by the zeros and ones that they can or cannot send. Thus, user "Bob" is really defined only by his ability to compute things that non-Bobs cannot compute. Since everybody can buy the same computers, Bob can be Bob only by his knowledge of some value that only Bob knows.
In the raw Diffie-Hellman exchange that you present, you talk to some entity that is supposed to generate a random secret value on-the-fly, and use that. Everybody can do such random generation. At no place in the protocol is there any operation that only a specific Bob can do. Thus, the protocol cannot achieve any kind of authentication -- you don't know who you are talking to. Without authentication, impersonation is feasible, and that includes simultaneous double impersonation, better known as Man-in-the-Middle. At best, raw Diffie-Hellman provides a weaker feature: though you do not know who you are talking to, you still know that you are talking to the same entity throughout the session.
A single cryptographic algorithm won't get you far; any significant communication protocol will assemble several algorithms so that some definite security characteristics are achieved. A prime example is SSL/TLS; another is SSH. In SSH, a Diffie-Hellman key exchange is used, but the server's public part (its gb mod p) is signed by the server. The client knows that it talks to the right server because the client remembers (from a previous initialization step) the server's public key (usually of type RSA or DSA); in the model explained above, the rightful servers is defined and distinguished from imitators by its knowledge of the signature private key corresponding to the public key remembered by the client. That signature provides the authentication; the Diffie-Hellman then produces a shared secret that will be used to encrypt and protect all the data exchanges for that connection (using some symmetric encryption and MAC algorithms).
Thus, while Diffie-Hellman does not do everything you need by itself, it still provides a useful feature, namely a key exchange, that you would not obtain from digital signatures, and that provides the temporary shared secret needed to encrypt the actually exchanged data.
