Despite the similarity in spelling for today’s words, they have very different meanings.

contemn: [verb] scorn, disregard, feel contempt for
He seems to take price in his ability to contemn new performers. 

contemplate: [verb] consider, envisage and observe consistently about a likely or probably action or outcome
A wise person will contemplate all options before making an important decision.

The word contemn is not used a lot, but some people question whether it was the intended word in line two of The Ode (part of For The Fallen by  Laurence Binyon and used in ANZAC and other remembrance services around the world):

They shall grow not old, as we that are left grow old;
Age shall not weary them, nor the years condemn.
At the going down of the sun and in the morning
We will remember them.

It appears that Binyon did intend to use the word ‘condemn’, but ‘contemn’ would also fit in there. Which word gives the better message to you?