Can trust be re-born in a marriage? Yes, if integrity is re-born. Trust dies when deceit is born. When you lie to your spouse in order to cover-up some behavior, you have taken the first step in killing trust. A few more lies and trust will be destroyed. The only way to restore trust is to confess your sin, ask for mercy, and then re-commit yourself to telling the truth.
On the practical level this means that you must establish a new record of being trustworthy. Invite your spouse to investigate your behavior. Every time your mate discovers your actions matching your words, trust grows. It takes time, but you can become a person of integrity and your spouse can come again to trust you.