diff --git a/compiler/rustc_typeck/src/impl_wf_check/min_specialization.rs b/compiler/rustc_typeck/src/impl_wf_check/min_specialization.rs
index 2741d9f776ce2..5bebd7dee09b9 100644
--- a/compiler/rustc_typeck/src/impl_wf_check/min_specialization.rs
+++ b/compiler/rustc_typeck/src/impl_wf_check/min_specialization.rs
@@ -423,13 +423,10 @@ fn trait_predicate_kind<'tcx>(
     predicate: ty::Predicate<'tcx>,
 ) -> Option<TraitSpecializationKind> {
     match predicate.kind().skip_binder() {
-        ty::PredicateKind::Trait(ty::TraitPredicate {
-            trait_ref,
-            constness: ty::BoundConstness::NotConst,
-            polarity: _,
-        }) => Some(tcx.trait_def(trait_ref.def_id).specialization_kind),
-        ty::PredicateKind::Trait(_)
-        | ty::PredicateKind::RegionOutlives(_)
+        ty::PredicateKind::Trait(ty::TraitPredicate { trait_ref, constness: _, polarity: _ }) => {
+            Some(tcx.trait_def(trait_ref.def_id).specialization_kind)
+        }
+        ty::PredicateKind::RegionOutlives(_)
         | ty::PredicateKind::TypeOutlives(_)
         | ty::PredicateKind::Projection(_)
         | ty::PredicateKind::WellFormed(_)
diff --git a/src/test/ui/rfc-2632-const-trait-impl/specializing-constness-2.rs b/src/test/ui/rfc-2632-const-trait-impl/specializing-constness-2.rs
new file mode 100644
index 0000000000000..633543700d2ec
--- /dev/null
+++ b/src/test/ui/rfc-2632-const-trait-impl/specializing-constness-2.rs
@@ -0,0 +1,31 @@
+#![feature(const_trait_impl, min_specialization, rustc_attrs)]
+
+#[rustc_specialization_trait]
+#[const_trait]
+pub trait Sup {}
+
+impl const Sup for () {}
+
+#[const_trait]
+pub trait A {
+    fn a() -> u32;
+}
+
+impl<T: Default> A for T {
+    default fn a() -> u32 {
+        2
+    }
+}
+
+impl<T: Default + ~const Sup> const A for T {
+    fn a() -> u32 {
+        3
+    }
+}
+
+const fn generic<T: Default>() {
+    <T as A>::a();
+    //~^ ERROR: the trait bound `T: ~const Sup` is not satisfied
+}
+
+fn main() {}
diff --git a/src/test/ui/rfc-2632-const-trait-impl/specializing-constness-2.stderr b/src/test/ui/rfc-2632-const-trait-impl/specializing-constness-2.stderr
new file mode 100644
index 0000000000000..c554671e18d82
--- /dev/null
+++ b/src/test/ui/rfc-2632-const-trait-impl/specializing-constness-2.stderr
@@ -0,0 +1,19 @@
+error[E0277]: the trait bound `T: ~const Sup` is not satisfied
+  --> $DIR/specializing-constness-2.rs:27:5
+   |
+LL |     <T as A>::a();
+   |     ^^^^^^^^^^^^^ the trait `~const Sup` is not implemented for `T`
+   |
+note: required for `T` to implement `~const A`
+  --> $DIR/specializing-constness-2.rs:20:37
+   |
+LL | impl<T: Default + ~const Sup> const A for T {
+   |                                     ^     ^
+help: consider further restricting this bound
+   |
+LL | const fn generic<T: Default + ~const Sup>() {
+   |                             ++++++++++++
+
+error: aborting due to previous error
+
+For more information about this error, try `rustc --explain E0277`.
diff --git a/src/test/ui/rfc-2632-const-trait-impl/specializing-constness.rs b/src/test/ui/rfc-2632-const-trait-impl/specializing-constness.rs
new file mode 100644
index 0000000000000..ff0cd489d4744
--- /dev/null
+++ b/src/test/ui/rfc-2632-const-trait-impl/specializing-constness.rs
@@ -0,0 +1,26 @@
+#![feature(const_trait_impl, min_specialization, rustc_attrs)]
+
+#[rustc_specialization_trait]
+#[const_trait]
+pub trait Sup {}
+
+impl const Sup for () {}
+
+#[const_trait]
+pub trait A {
+    fn a() -> u32;
+}
+
+impl<T: ~const Default> const A for T {
+    default fn a() -> u32 {
+        2
+    }
+}
+
+impl<T: Default + Sup> A for T { //~ ERROR: cannot specialize
+    fn a() -> u32 {
+        3
+    }
+}
+
+fn main() {}
diff --git a/src/test/ui/rfc-2632-const-trait-impl/specializing-constness.stderr b/src/test/ui/rfc-2632-const-trait-impl/specializing-constness.stderr
new file mode 100644
index 0000000000000..3296c109c4e73
--- /dev/null
+++ b/src/test/ui/rfc-2632-const-trait-impl/specializing-constness.stderr
@@ -0,0 +1,8 @@
+error: cannot specialize on trait `Default`
+  --> $DIR/specializing-constness.rs:20:9
+   |
+LL | impl<T: Default + Sup> A for T {
+   |         ^^^^^^^
+
+error: aborting due to previous error
+
diff --git a/src/test/ui/specialization/const_trait_impl.rs b/src/test/ui/specialization/const_trait_impl.rs
new file mode 100644
index 0000000000000..05ba4c8d45d5b
--- /dev/null
+++ b/src/test/ui/specialization/const_trait_impl.rs
@@ -0,0 +1,55 @@
+// check-pass
+#![feature(const_trait_impl, min_specialization, rustc_attrs)]
+
+#[rustc_specialization_trait]
+#[const_trait]
+pub unsafe trait Sup {
+    fn foo() -> u32;
+}
+
+#[rustc_specialization_trait]
+#[const_trait]
+pub unsafe trait Sub: ~const Sup {}
+
+unsafe impl const Sup for u8 {
+    default fn foo() -> u32 {
+        1
+    }
+}
+
+unsafe impl const Sup for () {
+    fn foo() -> u32 {
+        42
+    }
+}
+
+unsafe impl const Sub for () {}
+
+#[const_trait]
+pub trait A {
+    fn a() -> u32;
+}
+
+impl<T: ~const Default> const A for T {
+    default fn a() -> u32 {
+        2
+    }
+}
+
+impl<T: ~const Default + ~const Sup> const A for T {
+    default fn a() -> u32 {
+        3
+    }
+}
+
+impl<T: ~const Default + ~const Sub> const A for T {
+    fn a() -> u32 {
+        T::foo()
+    }
+}
+
+const _: () = assert!(<()>::a() == 42);
+const _: () = assert!(<u8>::a() == 3);
+const _: () = assert!(<u16>::a() == 2);
+
+fn main() {}