Assuming that there is nothing like a previous, putative marriage that needs to be taken care of first (through a decree of nullity), and assuming that you both still have valid matrimonial consent, your marriage can be rendered valid using a canon law procedure known as radical sanation.
This term comes from the Latin phrase sanatio in radice, meaning “healing in the root.” According to the Code of Canon Law, “The radical sanation of an invalid marriage is its convalidation without the renewal of consent” (CIC 1161:1). This means you do not have to go through a new marriage ceremony.
For a radical sanation to take place, several conditions must apply. First and most basically, “A radical sanation is not to be granted unless it is probable that the parties intend to persevere in conjugal life” (CIC 1161:3). If there is evidence the one or both of the parties intends anything less than a permanent marriage, radical sanation is ruled out.
Second, “A marriage cannot be radically sanated if consent is lacking in either or both of the parties” (CIC 1162:1). You and your spouse must have valid consent regarding your marriage, and this consent must exist simultaneously in the two of you. At some point you must have consented freely to the marriage in a way that did not exclude any of the essential properties of marriage (monogamy, fidelity, permanence, and openness to children). This consent is presumed to have been given in your marriage ceremony outside the Church unless there is evidence otherwise (CIC 1107), and the consent is presumed to exist at the present unless one party has indicated otherwise.
Third, any impediments that exist must be taken care of. Many of these can be resolved as part of the radical sanation itself. In general, “A marriage which is invalid due to an impediment or due to defect of legitimate form can be sanated provided the consent of each party continues to exist” (CIC 1163:1). This would apply in your case because your marriage was invalid due to a defect of form (you failed to get a dispensation for a marriage ceremony outside the Church).
Some impediments cannot be dispensed in this manner: “A marriage which is invalid due to an impediment of the natural law or of divine positive law can be sanated only after the impediment has ceased to exist” (CIC 1163:2). Examples of such impediments include having a previous marriage bond or total, permanent impotence (which is different from sterility). The first example can cease to exist if the previous spouse is dead or if one has obtained a decree of nullity to show that there never was a valid marriage in the first place.
If your spouse would have an extremely bad reaction to the sanation procedure, then, for the sake of domestic peace, he would not need to be told about it: “A sanation can be granted validly even when one or both of the parties are unaware of it, but it is not to be granted except for serious reason” (CIC 1164). The extreme reaction of your spouse could count as the serious reason needed for this.
Normally your local bishop would be the one granting the sanation: “In individual cases radical sanation can be granted by the diocesan bishop, even if several reasons for nullity exist in the same marriage, provided the conditions mentioned in canon 1125 concerning the sanation of a mixed marriage are fulfilled” (CIC 1165:2).
Chief among the latter is the condition that “the Catholic party is to declare that he or she is prepared to remove dangers of defecting from the faith and is to make a sincere promise to do all in his or her power in order that all the children be baptized and brought up in the Catholic Church” (CIC 1125). In other words, you must promise to remain a Catholic and to do what you can to see that your children will be Catholics.
Call your parish priest or the marriage tribunal at your diocese to investigate obtaining a radical sanation.