The relationship between Law and Gospel is a fierce perennial debate, especially between the Lutheran and Reformed Churches. The Lutheran doctrine of Law and Gospel tends towards dualism and the Reformed tends towards identity, and although the Lutheran approach has had wider acceptance (even by much of the Reformed Churches), the Law and Gospel debate is far from reaching an unanimous resolution.
Karl Barth proposes a new solution to the Law and Gospel debate. Barth says that the Gospel is Law, and sides with the Reformed strategy of identifying Law and Gospel (contra the Lutherans). However, Barth is not merely repeating the old Reformed solution (Barth hardly ever repeats anything without modification!) In this post, I'll share Barth's solution to the Law and Gospel debate, and provide a helpful explanation of it by John D. Godsey.
Karl Barth's solution: Gospel is Law
Barth identifies Law and Gospel such that the Law is the result of the Gospel: Gospel is Law! I was immediately impressed by Barth's solution to the Law and Gospel question when I first read it in the Church Dogmatics, Vol II/2, because Barth evades the pitfalls of both the Lutheran and Reformed solutions. When Barth said the "Gospel itself has the form and fashion of the Law", he means that the Law is living out the Gospel, not returning to the Mosaic Law. Barth is not setting aside the Old Testament Law, like the critics of Lutherans claim, nor is Barth conflating or replacing the Gospel with the Old Testament Law, as critics of the Reformed theologians claim; Barth is maintaining the unity of Law and Gospel, without separation or division (like the words of Chalcedon).
There can be no question, therefore, of having to speak of anything other than the Gospel. . . . Ruling grace is commanding grace. The Gospel itself has the form and fashion of the Law. The one Word of God is both Gospel and Law. It is not Law by itself and independent of the Gospel. But it is also not Gospel without Law. In its content, it is Gospel; in its form and fashion, it is Law. It is first Gospel and then Law. It is the Gospel which contains and encloses the Law as the ark of the covenant the tables of Sinai. But it is both Gospel and Law. The one Word of God which is the revelation and work of His grace is also Law. 
John D. Godsey explains Barth's Gospel is Law
Barth's Gospel is Law is an elegant solution to the Law and Gospel debate, yet it is nuanced enough that it is difficult to further explain in plain language. So I was pleased to find a short and helpful summary of Barth's theology of Law and Gospel in the introduction of John D. Godsey's Karl Barth's Table Talk:
Directly following the Doctrine of God's Gracious Election, we have in Chapter 8 a discussion of God's Command. This is by no means an accident. Election and command belong inextricably together and must be viewed in that order. The correspondence to Professor Barth's famous coupling of 'Gospel and Law' over against the Lutheran 'Law and Gospel' is obvious. The two are to be differentiated, but never separated, and Law is always to be seen in the light of the Gospel. In Christian theological thinking it is of critical importance that we always move from Gospel to Law, just as we must go from justification to sanctification, from faith to works, from Church to State. In other words, here we have a Christological concept of the Law that excludes any foundation in natural law or orders of creation: God's nomos and God's logos are ultimately identical, for Law is the form of the Gospel. There is no Law and no Gospel in and for themselves, but only the one Word of God. This means that God's command is an event, not a general proposition. In our context it also means that the concrete form of election is sanctification. Jesus Christ is the sanctifying God and the sanctified man. 
—John D. Godsey
Karl Barth's Gospel is Law is a brilliant solution to the Law and Gospel debate that has divided Lutheran and Reformed Churches since the beginning of the Reformation. Barth's elegant Gospel is Law requires further elaboration in practice, and John D. Godsey's explanation (quoted above) is a handy explanation in plain language of what Barth means by Gospel is Law.
To further understand Barth's the Gospel is Law solution, I recommend reading the first paragraph of §36 "Ethics as a Task of the Doctrine of God" Church Dogmatics, Vol. II/2, and to understand how Barth's interpretation of the Old Testament Law, read Barth's assessment of J. Coccejus' Federal Theology from the Church Dogmatics, Vol. IV/1, in §57 "The Word of God the Reconciler".
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[^1] Barth, Karl. Church Dogmatics, Vol 2.2: The Doctrine of God. Trans. G. W. Bromiley and T. F. Torrance. Vol. 12. London: T & T Clark, 2010. 7. Print. Study Edition. 
[^2] Barth, Karl. Karl Barth's Table Talk. Ed. John D. Godsey. London: Oliver and Boyd, 1965. 7. Print.
Related: Federal Theology, Gospel is Law, Johannes Coccejus, John D. Godsey, Karl Barth, Law and Gospel, Mosaic Law, Old Testament Law