diff --git a/src/builder/dataflow.rs b/src/builder/dataflow.rs
index 1b547172e..24e580a2a 100644
--- a/src/builder/dataflow.rs
+++ b/src/builder/dataflow.rs
@@ -97,7 +97,8 @@ impl DFGBuilder<Hugr> {
 
 impl HugrBuilder for DFGBuilder<Hugr> {
     fn finish_hugr(mut self) -> Result<Hugr, ValidationError> {
-        self.base.infer_extensions()?;
+        let closure = self.base.infer_extensions()?;
+        self.base.validate_with_extension_closure(closure)?;
         Ok(self.base)
     }
 }
diff --git a/src/extension/infer.rs b/src/extension/infer.rs
index 07014b561..c8e707eb3 100644
--- a/src/extension/infer.rs
+++ b/src/extension/infer.rs
@@ -28,9 +28,23 @@ use thiserror::Error;
 pub type ExtensionSolution = HashMap<(Node, Direction), ExtensionSet>;
 
 /// Infer extensions for a hugr. This is the main API exposed by this module
-pub fn infer_extensions(hugr: &impl HugrView) -> Result<ExtensionSolution, InferExtensionError> {
+///
+/// Return a tuple of the solutions found for locations on the graph, and a
+/// closure: a solution which would be valid if all of the variables in the graph
+/// were instantiated to an empty extension set. This is used (by validation) to
+/// concretise the extension requirements of the whole hugr.
+pub fn infer_extensions(
+    hugr: &impl HugrView,
+) -> Result<(ExtensionSolution, ExtensionSolution), InferExtensionError> {
     let mut ctx = UnificationContext::new(hugr);
-    ctx.main_loop()
+    let solution = ctx.main_loop()?;
+    ctx.instantiate_variables();
+    let closed_solution = ctx.main_loop()?;
+    let closure: HashMap<(Node, Direction), ExtensionSet> = closed_solution
+        .into_iter()
+        .filter(|(loc, _)| !solution.contains_key(loc))
+        .collect();
+    Ok((solution, closure))
 }
 
 /// Metavariables don't need much
@@ -599,6 +613,16 @@ impl UnificationContext {
         }
         self.results()
     }
+
+    /// Instantiate all variables in the graph with the empty extension set.
+    /// This is done to solve metas which depend on variables, which allows
+    /// us to come up with a fully concrete solution to pass into validation.
+    pub fn instantiate_variables(&mut self) {
+        for m in self.variables.clone().into_iter() {
+            self.add_solution(m, ExtensionSet::new());
+        }
+        self.variables = HashSet::new();
+    }
 }
 
 #[cfg(test)]
@@ -856,7 +880,8 @@ mod test {
         let hugr = builder.base;
         // TODO: when we put new extensions onto the graph after inference, we
         // can call `finish_hugr` and just look at the graph
-        let solution = infer_extensions(&hugr)?;
+        let (solution, extra) = infer_extensions(&hugr)?;
+        assert!(extra.is_empty());
         assert_eq!(
             *solution.get(&(src.node(), Direction::Outgoing)).unwrap(),
             rs
diff --git a/src/extension/validate.rs b/src/extension/validate.rs
index 55cc38d08..732f76e0d 100644
--- a/src/extension/validate.rs
+++ b/src/extension/validate.rs
@@ -20,9 +20,12 @@ pub struct ExtensionValidator {
 impl ExtensionValidator {
     /// Initialise a new extension validator, pre-computing the extension
     /// requirements for each node in the Hugr.
-    pub fn new(hugr: &Hugr) -> Self {
+    ///
+    /// The `closure` argument is a set of extensions which doesn't actually
+    /// live on the graph, but is used to close the graph for validation
+    pub fn new(hugr: &Hugr, closure: HashMap<(Node, Direction), ExtensionSet>) -> Self {
         let mut validator = ExtensionValidator {
-            extensions: HashMap::new(),
+            extensions: closure,
         };
 
         for node in hugr.nodes() {
diff --git a/src/hugr.rs b/src/hugr.rs
index fc7906fa0..4e1c52edf 100644
--- a/src/hugr.rs
+++ b/src/hugr.rs
@@ -7,7 +7,7 @@ pub mod serialize;
 pub mod validate;
 pub mod views;
 
-use std::collections::VecDeque;
+use std::collections::{HashMap, VecDeque};
 use std::iter;
 
 pub(crate) use self::hugrmut::HugrInternalsMut;
@@ -194,10 +194,12 @@ impl Hugr {
     }
 
     /// Infer extension requirements
-    pub fn infer_extensions(&mut self) -> Result<(), InferExtensionError> {
-        let solution = infer_extensions(self)?;
+    pub fn infer_extensions(
+        &mut self,
+    ) -> Result<HashMap<(Node, Direction), ExtensionSet>, InferExtensionError> {
+        let (solution, extension_closure) = infer_extensions(self)?;
         self.instantiate_extensions(solution);
-        Ok(())
+        Ok(extension_closure)
     }
 
     /// TODO: Write this
diff --git a/src/hugr/validate.rs b/src/hugr/validate.rs
index f6521c31b..f063ec2fd 100644
--- a/src/hugr/validate.rs
+++ b/src/hugr/validate.rs
@@ -14,7 +14,7 @@ use pyo3::prelude::*;
 
 use crate::extension::{
     validate::{ExtensionError, ExtensionValidator},
-    InferExtensionError,
+    ExtensionSet, InferExtensionError,
 };
 use crate::ops::validate::{ChildrenEdgeData, ChildrenValidationError, EdgeValidationError};
 use crate::ops::{OpTag, OpTrait, OpType, ValidateOp};
@@ -38,20 +38,35 @@ struct ValidationContext<'a> {
 }
 
 impl Hugr {
-    /// Check the validity of the HUGR.
+    /// Check the validity of the HUGR, assuming that it has no open extension
+    /// variables.
+    /// TODO: Add a version of validation which allows for open extension
+    /// variables (see github issue #457)
     pub fn validate(&self) -> Result<(), ValidationError> {
-        let mut validator = ValidationContext::new(self);
+        self.validate_with_extension_closure(HashMap::new())
+    }
+
+    /// Check the validity of a hugr, taking an argument of a closure for the
+    /// free extension variables
+    pub fn validate_with_extension_closure(
+        &self,
+        closure: HashMap<(Node, Direction), ExtensionSet>,
+    ) -> Result<(), ValidationError> {
+        let mut validator = ValidationContext::new(self, closure);
         validator.validate()
     }
 }
 
 impl<'a> ValidationContext<'a> {
     /// Create a new validation context.
-    pub fn new(hugr: &'a Hugr) -> Self {
+    pub fn new(
+        hugr: &'a Hugr,
+        extension_closure: HashMap<(Node, Direction), ExtensionSet>,
+    ) -> Self {
         Self {
             hugr,
             dominators: HashMap::new(),
-            extension_validator: ExtensionValidator::new(hugr),
+            extension_validator: ExtensionValidator::new(hugr, extension_closure),
         }
     }