Firstly, you need to apply a padding scheme like PKCS#5 or PKCS#7 so that your message will be always a multiple of 128. A 128-bit message than will be two-block.
However, if you have one byte less than the block size, you will have 128-bit messages this means that you have one block.
After the padding, you can encrypt with CBC mode. The first block will be x-ored with the IV. In CBC mode the IV must be unpredictable.
If you have only messages with 128-bit, you may not need the padding. if you want, you can skip the padding, just encrypt the message.
Having always one full block shouldn't put you into the ECB mode trap. ECB mode insecure and once you send a message again, you start to reveal information. You need at least Probabilistic Encryption. The better is authenticated encryption modes like AES-GCM or ChaCha-Poly1305.