diff --git a/jaxlib/gpu/rnn.cc b/jaxlib/gpu/rnn.cc
index c88b164e6..eaa815d33 100644
--- a/jaxlib/gpu/rnn.cc
+++ b/jaxlib/gpu/rnn.cc
@@ -13,6 +13,8 @@ See the License for the specific language governing permissions and
 limitations under the License.
 ==============================================================================*/
 
+#include <cstddef>
+
 #include "nanobind/nanobind.h"
 #include "nanobind/stl/pair.h"
 #include "jaxlib/absl_status_casters.h"
@@ -29,7 +31,7 @@ namespace nb = nanobind;
 nb::bytes BuildRnnDescriptor(int input_size, int hidden_size, int num_layers,
                              int batch_size, int max_seq_length, float dropout,
                              bool bidirectional, bool cudnn_allow_tf32,
-                             int workspace_size, int reserve_space_size) {
+                             size_t workspace_size, size_t reserve_space_size) {
   return PackDescriptor(RnnDescriptor{
       input_size, hidden_size, num_layers, batch_size, max_seq_length, dropout,
       bidirectional, cudnn_allow_tf32, workspace_size, reserve_space_size});
diff --git a/jaxlib/gpu/rnn_kernels.cc b/jaxlib/gpu/rnn_kernels.cc
index 89a6d0a30..e9820bc31 100644
--- a/jaxlib/gpu/rnn_kernels.cc
+++ b/jaxlib/gpu/rnn_kernels.cc
@@ -15,6 +15,7 @@ limitations under the License.
 
 #include "jaxlib/gpu/rnn_kernels.h"
 
+#include <cstddef>
 #include <utility>
 #include <vector>
 
@@ -71,7 +72,7 @@ template <>
 
 namespace JAX_GPU_NAMESPACE {
 
-static absl::StatusOr<std::pair<int, int>>
+static absl::StatusOr<std::pair<size_t, size_t>>
 DoRnnComputeWorkspaceReserveSpaceSizes(int input_size, int hidden_size,
                                        int num_layers, int batch_size,
                                        int max_seq_length, float dropout,
@@ -174,7 +175,7 @@ DoRnnComputeWorkspaceReserveSpaceSizes(int input_size, int hidden_size,
   return std::make_pair(workSpaceSize, reserveSpaceSize);
 }
 
-absl::StatusOr<std::pair<int, int>> RnnComputeWorkspaceReserveSpaceSizes(
+absl::StatusOr<std::pair<size_t, size_t>> RnnComputeWorkspaceReserveSpaceSizes(
     int input_size, int hidden_size, int num_layers, int batch_size,
     int max_seq_length, float dropout, bool bidirectional,
     bool cudnn_allow_tf32) {
diff --git a/jaxlib/gpu/rnn_kernels.h b/jaxlib/gpu/rnn_kernels.h
index 468c02eac..e95b77883 100644
--- a/jaxlib/gpu/rnn_kernels.h
+++ b/jaxlib/gpu/rnn_kernels.h
@@ -16,6 +16,8 @@ limitations under the License.
 #ifndef JAXLIB_GPU_RNN_KERNELS_H_
 #define JAXLIB_GPU_RNN_KERNELS_H_
 
+#include <cstddef>
+
 #include "absl/status/statusor.h"
 #include "jaxlib/gpu/vendor.h"
 #include "xla/ffi/api/ffi.h"
@@ -34,12 +36,12 @@ struct RnnDescriptor {
   float dropout;
   int bidirectional;
   int cudnn_allow_tf32;
-  int workspace_size;
-  int reserve_space_size;
+  size_t workspace_size;
+  size_t reserve_space_size;
 };
 
 // Return (workspace size, reserve space size).
-absl::StatusOr<std::pair<int, int>> RnnComputeWorkspaceReserveSpaceSizes(
+absl::StatusOr<std::pair<size_t, size_t>> RnnComputeWorkspaceReserveSpaceSizes(
     int input_size, int hidden_size, int num_layers, int batch_size,
     int max_seq_length, float dropout, bool bidirectional,
     bool cudnn_allow_tf32);
diff --git a/tests/experimental_rnn_test.py b/tests/experimental_rnn_test.py
index 376a9b1a1..7fa3b93f3 100644
--- a/tests/experimental_rnn_test.py
+++ b/tests/experimental_rnn_test.py
@@ -213,8 +213,36 @@ class RnnTest(jtu.JaxTestCase):
 
     k = jax.random.split(jax.random.PRNGKey(1), 4)
     stablehlo = jax.jit(f).lower(*k).as_text("stablehlo")
-    self.assertIn('"\\01\\00\\00\\00\\01\\00\\00\\00\\01\\00\\00\\00\\01\\00\\00\\00\\01\\00\\00\\00\\00\\00\\00\\00\\00\\00\\00\\00\\01\\00\\00\\00@\\03\\80\\00@\\01\\00\\00"',
-                  stablehlo)
+    if jtu.jaxlib_version() <= (0, 5, 2):
+      self.assertIn('"\\01\\00\\00\\00\\01\\00\\00\\00\\01\\00\\00\\00\\01\\00\\00\\00\\01\\00\\00\\00\\00\\00\\00\\00\\00\\00\\00\\00\\01\\00\\00\\00@\\03\\80\\00@\\01\\00\\00"',
+                    stablehlo)
+    else:
+      self.assertIn('"\\01\\00\\00\\00\\01\\00\\00\\00\\01\\00\\00\\00\\01\\00\\00\\00\\01\\00\\00\\00\\00\\00\\00\\00\\00\\00\\00\\00\\01\\00\\00\\00@\\03\\80\\00\\00\\00\\00\\00@\\01\\00\\00\\00\\00\\00\\00"',
+                    stablehlo)
+
+  @jtu.run_on_devices("cuda")
+  def test_no_workspace_overflow(self):
+    if jtu.jaxlib_version() <= (0, 5, 2):
+      self.skipTest("Older versions fail because of integer overflow.")
+
+    # Problem sizes known to cause overflows on older versions.
+    batch_size, max_seq_length, input_size = 256, 500, 512
+    num_layers, hidden_size = 1, 256
+    num_params = rnn.get_num_params_in_lstm(
+        input_size, hidden_size, num_layers, True)
+    x = jax.ShapeDtypeStruct(
+        (batch_size, max_seq_length, input_size), jnp.float32)
+    h_0 = jax.ShapeDtypeStruct(
+        (2 * num_layers, batch_size, hidden_size), jnp.float32)
+    c_0 = jax.ShapeDtypeStruct(
+        (2 * num_layers, batch_size, hidden_size), jnp.float32)
+    weights = jax.ShapeDtypeStruct((num_params,), jnp.float32)
+    seq_lengths = jax.ShapeDtypeStruct((batch_size,), jnp.int32)
+    fun = jax.jit(partial(
+        rnn.lstm, input_size=input_size, hidden_size=hidden_size,
+        num_layers=num_layers, dropout=0.0, bidirectional=True))
+    fun.lower(x, h_0, c_0, weights, seq_lengths)  # Doesn't crash.
+
 
 if __name__ == '__main__':
   absltest.main(testLoader=jtu.JaxTestLoader())