Computer Science is far more than programming! In fact, I have been told that the way to tell a good "Computer Science" or "Computer Engineering" program from a bad one is that the good program will NOT have any courses titled or specifically about programming in specific languages. You are expected to pick up the languages while learning such things as (from the M.I.T. computer science website)
Dynamic Programming
Principals of Computer Science
Discrete Stochastic Processes
Neural Nets
and more.
The fact is that most science majors know how to program in at least one language and can pick up others easily. Asking if you can get a job in a computer field if you can program is a lot like asking if you can get a job in linguistics if you speak one language!
(Yes, there are people in computer fields, even without a college degree, who do nothing but program. That is a very low level, low paid, job, not much above "data entry". It is the people who decide what programs should be written that do the real "computer" work and get paid better.)
If you like mathematics enough more than computer science to consider majoring in mathematics rather than computer sciences, why are you talking about getting a job programming rather than in mathematics?