Encrypting a 180-bit plaintext into a 180 bit ciphertext with a 128-bit block cipher
If you have a unique, unchanging identifier for each entry in your data store, you can use counter mode.

A nice thing about counter mode turns a block cipher into a stream cipher. No matter what the block size is, CTR mode encrypts an $n$-bit plaintext into an $n$-bit ciphertext.

In order to achieve that, CTR requires a unique counter value per block. Note: not just a unique counter value per message, but a unique counter value per block. The counter size is the same as the block size. In your case, you have messages that fit on two blocks, thus each message requires two counter values. If you have a unique identifier $k$ for each message, you can use $k$ and $k+1$ as the counter values for the two blocks (the second of which is partial) of the message.

Thus you need a 127-bit unique identifier for each message (128-bit block, minus one bit to distinguish the two blocks inside each message). The only security requirement for these 127 bits is that they are never reused for a given key. The initial counter value to encrypt a message is often chosen randomly, but this is not a requirement, just a convenience to ensure uniqueness. Of course, to decrypt the data, you need to be able to recover the unique identifier associated with each entry.

If your entries have some kind of unique identifier, which is often the case in databases, then you're set. Just remember that if you move data around or normalize it in a way that changes the identifiers, you will need to decrypt and reencrypt the data.

Some crypto libraries may present CTR mode through a function that randomly generates the initial counter value and prepends it to the message (so you'd input a 160-bit plaintext and get back a 288-bit ciphertext). Use a library that lets you specify the initial counter value (almost all implementations will increment the counter by 1 for each successive block, so pick initial counter values that are even, but you'll need to be aware of the endianness used by your library).

Keep in mind that encryption only gives you confidentiality, not integrity. In other words, someone who obtains the ciphertexts but not the key will not be able to find any information about the data; but if someone can inject fake ciphertexts or modify existing ciphertexts, the tampering cannot be detected. It is intrinsically impossible to detect tampering by cryptographic means in your scenario since there is no room for any redundancy.