CBC mode with predictable IV ok if key is only used once?

Someone told me recently that using CBC mode with a predictable (e.g. all 0) IV is reasonably secure if a key is only used for encryption one time. I've gone through a couple of examples of chosen plaintext attacks against CBC mode with a predictable IV and it does seem that they assume that the key that is used for the attacker's encrypted text is the same as the key used for the attackee's encrypted text. Also, in this instance, an attacker could not control the entirety of the plain text that is encrypted, just a small portion of it.

So, is this person correct, does using the key for encryption only once make using a predictable IV safe or is there an issue I'm not seeing?

So, is this person correct, does using the key for encryption only once make using a predictable IV safe?

Yes, he is correct.

This attack works by the attacker obtaining a CBC-mode ciphertext, and wants to verify the plaintext contents of a specific plaintext block. To do that, he computes that the input to the block cipher would be if that guess was correct (and the corresponding output), and based on that, create a plaintext message that, with the predictable IV, send that input to the block cipher. He then asks for that plaintext message to be encrypted with the same key. If the corresponding ciphertext message has the expected output block, he then knows that his guess was correct.

If we encrypt only a single message, then he cannot perform this attack - he can learn the original ciphertext, but then he cannot ask for a second message to be encrypted with the same key.

Now, this holds if the encryptor takes the entire plaintext, and encrypts it in its entirety, and that full ciphertext is sent to the receiver (and the adversary). If you run CBC mode incrementally, for example, you take the first part of the plaintext, encrypt it, send it out, and then take the second part plaintext, and then encrypt that, then that's not safe - if the attacker can listen to the first part of the ciphertext, and then (based on that) modify what the second part of the plaintext, he can implement the 'predictable IV' attack - even though it may not look like you're sending a second IV, because of how CBC mode works, you effectively are.

Also, in this instance, an attacker could not control the entirety of the plain text that is encrypted, just a small portion of it.

Doesn't matter - it's still safe even if the attacker has arbitrary control. For example, if the adversary can specify the entire message except for one bit, he still can't learn what that one bit is.

So, is this person correct, does using the key for encryption only once make using a predictable IV safe?

No, he is incorrect at least academically. And in some cases practically.

That's because of so-called multi-target attacks. Assuming $b$-bit keys, and $k$ random keys are used (and thus $k$ ciphertexts) for known common (or chosen) first block $P_0$ of plaintext (e.g. same start of the payload), there is a so-called multi-target attack with expected cost $2^{b-1}/k$ encryptions recovering one full plaintext, that is $k$ times faster than with random IV. That attack simply enumerate keys, enciphers the known fixed $\text{IV}\oplus P_0$, and searches the outcome in a table of all first ciphertext blocks (which can be optimized to cost just over a single memory access).

Even when the "known common (or chosen) first block of plaintext" condition is met, this attack is seldom a disaster in practice for random 128-bit or larger keys, even for very large $k$ (which would be many millions in some realistic scenarios, e.g. session keys). That's because storing and accessing the necessary table actually has a non-negligible cost (investment and power). It's hard to imagine we loose more than 20 bits of security against an attacker using ASICs, which is the most reasonable way to carry such attack against a 128-bit key by brute force. But the problem could be devastating if the key was deterministically derived from a password, and there's even a thousand passwords used.