Misdemeanor and Employment

My question involves criminal records for the state of: California
Hi there,
I need your help. I just got a job offer from a government agency. They are going to run a background check on me. I had convicted a misdemeanor last year (shoplifting), and I am done with my probation. I am in the process of having my record expunged.
The background check will be done after I start the job. I know my record will show up. My question is, is it possible they will terminate my employment? After they interviewed me and officially offered me the job?
I wish they run a background check before offering it to me, so, if it is not okay, I still have my current job.
When I applied the job, I did not commit it yet. The process of hiring take years. On the application that I applied, I did answer No for conviction, because I did not convict at that time.
Now, they offer me the job and did not ask me to submit a new application.
And if they do, I plan to answer truthfully about my conviction.
It is a professional job and in deed my dream job, but I don't want to leave my current job, accept the offer, then, got terminate when they run background check.
it is my very first offense, and I am an adult. I made a stupidiest mistake of my life, and I am so ashamed about it. I just hope I can move on with my life and start it over again with this job.
sorry for a long post.

You lied on application that alone is grounds for termination

I did not lie at all. At the time of applying, I was not commit any crime.
The incident was last year, Sept time frame. The job I applied was March time frame before the incident.

You might alert them to this. The conviction could keep your from government job. Also they might "assume" you lied and terminate you. Either way its best to be honest and upfront regardless of outcome