diff --git a/jax/_src/lax/lax.py b/jax/_src/lax/lax.py
index 4a2887045fb1..b4a4568f08b9 100644
--- a/jax/_src/lax/lax.py
+++ b/jax/_src/lax/lax.py
@@ -428,8 +428,8 @@ def convert_element_type(operand: Array, new_dtype: DType = None,
   if hasattr(operand, '__jax_array__'):
     operand = operand.__jax_array__()
 
-  # Note: don't canonicalize old_dtype because x64 context might
-  # cause un-canonicalized operands to be passed in.
+  # Don't canonicalize old_dtype because x64 context might cause
+  # un-canonicalized operands to be passed in.
   old_dtype = np.result_type(operand)
   old_weak_type = dtypes.is_weakly_typed(operand)
 
@@ -441,6 +441,14 @@ def convert_element_type(operand: Array, new_dtype: DType = None,
     msg = "Casting complex values to real discards the imaginary part"
     warnings.warn(msg, np.ComplexWarning, stacklevel=2)
 
+  # Python has big integers, but convert_element_type(2 ** 100, np.float32) need
+  # not be an error since the target dtype fits the value. Handle this case by
+  # converting to a NumPy array before calling bind. Without this step, we'd
+  # first canonicalize the input to a value of dtype int32 or int64, leading to
+  # an overflow error.
+  if type(operand) is int:
+    operand = np.asarray(operand, new_dtype)
+
   if ((old_dtype, old_weak_type) == (new_dtype, new_weak_type)
       and isinstance(operand, (core.Tracer, xla.DeviceArray))):
     return operand
diff --git a/jax/abstract_arrays.py b/jax/abstract_arrays.py
index 734957cdd1b0..57c75a0774c8 100644
--- a/jax/abstract_arrays.py
+++ b/jax/abstract_arrays.py
@@ -68,9 +68,8 @@ def _zeros_like_python_scalar(t, x):
   return np.array(0, dtypes.python_scalar_dtypes[t])
 
 def _make_concrete_python_scalar(t, x):
-  return ConcreteArray(
-    np.array(x, dtype=dtypes.python_scalar_dtypes[t]),
-    weak_type=True)
+  return ConcreteArray(np.array(x, dtype=dtypes.python_scalar_dtypes[t]),
+                       weak_type=True)
 
 for t in dtypes.python_scalar_dtypes:
   core.pytype_aval_mappings[t] = partial(_make_concrete_python_scalar, t)
diff --git a/tests/api_test.py b/tests/api_test.py
index 4b38afd46111..3bf5a9c31852 100644
--- a/tests/api_test.py
+++ b/tests/api_test.py
@@ -2393,6 +2393,12 @@ def f(_):
     expected = jnp.arange(1) + 1
     self.assertAllClose(ans, expected)
 
+  def test_large_python_int_to_float(self):
+    # https://github.com/google/jax/pull/6165
+    jnp.multiply(2 ** 100, 3.)  # doesn't crash
+    out = lax.convert_element_type(2 ** 100, jnp.float32)  # doesn't crash
+    self.assertArraysEqual(out, np.float32(2 ** 100))
+
 
 class RematTest(jtu.JaxTestCase):
 
diff --git a/tests/random_test.py b/tests/random_test.py
index 41fa226bc71a..3eb3e6b54bf5 100644
--- a/tests/random_test.py
+++ b/tests/random_test.py
@@ -746,6 +746,8 @@ def f(x):
     grad(lambda x: jnp.sum(vmap(f)(x)))(jnp.ones(2))
 
   def testNoOpByOpUnderHash(self):
+    if not config.omnistaging_enabled:
+      raise SkipTest("test requires omnistaging")
     def fail(*args, **kwargs): assert False
     apply_primitive, xla.apply_primitive = xla.apply_primitive, fail
     try: