Recover key given partial key, IV and cipher text using CBC mode Given a known IV and a small part of the key (not enough to use brute force)
is there are any way to break CBC encryption with only one cipher text?
A known IV doesn't help: it's meant to be public.

Even if you know a part of the key then there's still no attack that's easier than trying all possible keys. Knowing a part of the key merely makes the set of possible keys smaller.

Knowing even a large amount of plaintext and ciphertext doesn't help to recover the key. As long as you haven't fully found the key, having access to plaintext-ciphertext pairs doesn't help recover the key more quickly. And knowing part of the key does not help to generate new valid plaintext/ciphertext pairs.

So leaking a small part of the key does not weaken the system. The problem with leaking a small part of the key is that when this happens, it's usually difficult to ensure that only a small enough part of the key is leaked, where small enough means “so that my adversaries can't brute-force the key”.

Note that, as Luis Casillas points out, here I'm assuming that the underlying block cipher is an ideal cipher, which is a good enough assumption for block ciphers that are used in practice. (Not in theory though — related key attacks do exist on AES, for example, but they're theoretical: they still require an unfeasible amount of computation). See Luis Casillas's answer for a more correct explanation.