Correct spelling, explanation: assign originates in Latin assigno, and despite some changes in the ending of the word to make it sound more English, the double s stayed untouched as a link to the word’s roots. If you know the origins of the word or the same word in another language, you can notice that consonants usually stay the same.

Definition of assign:
verb
1. to give a job or a task to someone or something
We wanted to assign Katy to be the group leader, but some people were against it.
2. to prepare something to be used for a specific reason
We always have some spare money we assign for emergencies only.

Incorrect spelling, explanation: there is a set of adjectives in English constructed with an a- and a verb or noun (alight, asleep), and asign would look like one of them, which would completely change its meaning. The correct word is assign and it comes from Latin assigno, which also explains why it’s spelled with double s.