No, that is impossible. The reason is simple: How would you decrypt this? If you input the ciphertext and the key into the decryption function, than you have to get exactly one output, not two. How would you decide which output is the correct one?
The DES encryption and decryption functions are bijective under one given key. This means that for every possible input there is exactly one output. Not more, not less. Every output has only one matching input, not more, not less. You can reverse any transformation made, forward and backward, at least if you have the key. This behavior is assured by the inner Feistel network of DES.
All this has to be the case for every possible key, even weak or semi-weak ones.
This is the case for every "standard" block cipher: DES, AES, Blowfish, Twofish, MARS, Serpent, RC5, RC6, ... They all meet this requirement. Different operating modes can change this, through. Different plaintexts can lead to the same ciphertext with different IVs, for example for CBC. I don't think that the question focuses at this, but I want to include it after reading Maarten Bodewes - owlstead's answer.