diff --git a/frontend_test.go b/frontend_test.go
new file mode 100644
index 00000000..b82552bf
--- /dev/null
+++ b/frontend_test.go
@@ -0,0 +1,70 @@
+package pgconn_test
+
+import (
+	"context"
+	"io"
+	"os"
+	"testing"
+
+	"github.com/jackc/pgconn"
+	"github.com/jackc/pgproto3/v2"
+	"github.com/stretchr/testify/assert"
+	"github.com/stretchr/testify/require"
+)
+
+// frontendWrapper allows to hijack a regular frontend, and inject a specific response
+type frontendWrapper struct {
+	front pgconn.Frontend
+
+	msg pgproto3.BackendMessage
+}
+
+// frontendWrapper implements the pgconn.Frontend interface
+var _ pgconn.Frontend = (*frontendWrapper)(nil)
+
+func (f *frontendWrapper) Receive() (pgproto3.BackendMessage, error) {
+	if f.msg != nil {
+		return f.msg, nil
+	}
+
+	return f.front.Receive()
+}
+
+func TestFrontendFatalErrExec(t *testing.T) {
+	t.Parallel()
+
+	config, err := pgconn.ParseConfig(os.Getenv("PGX_TEST_CONN_STRING"))
+	require.NoError(t, err)
+
+	buildFrontend := config.BuildFrontend
+	var front *frontendWrapper
+
+	config.BuildFrontend = func(r io.Reader, w io.Writer) pgconn.Frontend {
+		wrapped := buildFrontend(r, w)
+		front = &frontendWrapper{wrapped, nil}
+
+		return front
+	}
+
+	conn, err := pgconn.ConnectConfig(context.Background(), config)
+	require.NoError(t, err)
+	require.NotNil(t, conn)
+	require.NotNil(t, front)
+
+	// set frontend to return a "FATAL" message on next call
+	front.msg = &pgproto3.ErrorResponse{Severity: "FATAL", Message: "unit testing fatal error"}
+
+	_, err = conn.Exec(context.Background(), "SELECT 1").ReadAll()
+	assert.Error(t, err)
+
+	err = conn.Close(context.Background())
+	assert.NoError(t, err)
+
+	select {
+	case <-conn.CleanupDone():
+		t.Log("ok, CleanupDone() is not blocking")
+
+	default:
+		assert.Fail(t, "connection closed but CleanupDone() still blocking")
+	}
+}