Is .NET DESCryptoServiceProvider secure in this case?
The key space for DES is far too small (56 bits). Therefore, any use of DES is not secure. It doesn't matter what mode you use. If the attacker has one plaintext, ciphertext pair, they can brute force the key space and recover the key in a feasible amount of time (24 hours using the cloud).


  But most importantly, how it could be made secure? Will change to AES help?


Moving to AES will definitely help. What will help the most is first, defining the desired protection model (i.e., what does "secure" mean in your context?). Do you only care about confidentiality? Do you also care about integrity? How about authenticity? Deciding this will help you decide which mode to use. In particular, do you need a MAC or authenticated encryption mode?

Once you have this figured out, choose the cipher and mode to match your needs, not just security needs, but also performance needs.