I like Scott's elliptic curve construction, but here is another construction of essentially the same thing with more explicit formulas. Let your discrete valuation ring be R = k[[b]] and your group scheme be Spec of the ring S = R[t,(1-bt)^-1]. Then Spec(S) is a group scheme using the multiplication rule

μ(t1,t2) = t1 + t2 - bt1t2.

and inverse

ν(t) = -t(1-bt)-1.

(Meaning, this is either a Hopf algebra structure or I view it as representing this group-valued functor on rings, depending on your theology.)

When b is invertible, then replacing t with the new coordinate s = 1-bt makes this isomorphic to the multiplicative group scheme (and, in fact, that's how the above formulas for multiplication and inversion are easiest to obtain).

When b is zero, this reduces to the additive group explicitly.

I like Scott's elliptic curve construction, but here is another construction of essentially the same thing with more explicit formulas. Let your discrete valuation ring be R = k[[b]] and your group scheme be Spec of the ring S = R[t,(1-bt)^-1]. Then Spec(S) is a group scheme using the multiplication rule

μ(t1,t2) = t1 + t2 - bt1t2.

and inverse

ν(t) = -t(1-bt)-1.

(Meaning, this is either a Hopf algebra structure or I view it as representing this group-valued functor on rings, depending on your theology.)

When b is invertible, then replacing t with the new coordinate s = 1-bt makes this isomorphic to the multiplicative group scheme (and, in fact, that's how the above formulas for multiplication and inversion are easiest to obtain).

When b is zero, this reduces to the additive group.