diff --git a/src/pyrecest/scenarios.py b/src/pyrecest/scenarios.py
index 62222f9c0..53a83e3c0 100644
--- a/src/pyrecest/scenarios.py
+++ b/src/pyrecest/scenarios.py
@@ -180,12 +180,16 @@ def _normalized_particle_weights(raw_weights: Any, particle_count: int, backend)
     if any(weight < 0.0 for weight in weight_values):
         raise ValueError("weights must be nonnegative")
 
-    weight_total = sum(weight_values)
-    if weight_total <= 0.0:
+    weight_scale = max(weight_values, default=0.0)
+    if weight_scale <= 0.0:
         raise ValueError("weights must have positive total mass")
+    scaled_weights = [weight / weight_scale for weight in weight_values]
+    scaled_total = sum(scaled_weights)
+    if scaled_total <= 0.0 or not math.isfinite(scaled_total):
+        raise ValueError("weights must have positive finite total mass")
 
     return backend.asarray(
-        [weight / weight_total for weight in weight_values],
+        [weight / scaled_total for weight in scaled_weights],
         dtype=backend.float64,
     )
 
diff --git a/tests/test_scenario_particle_weight_overflow.py b/tests/test_scenario_particle_weight_overflow.py
new file mode 100644
index 000000000..fd4eba407
--- /dev/null
+++ b/tests/test_scenario_particle_weight_overflow.py
@@ -0,0 +1,36 @@
+import math
+import sys
+from pathlib import Path
+
+from pyrecest.scenarios import run_scenario
+
+
+def test_particle_resampling_scenario_normalizes_extreme_finite_weights(
+    tmp_path: Path,
+):
+    max_float = sys.float_info.max
+    scenario = tmp_path / "extreme_particle_weights.toml"
+    scenario.write_text(
+        f"""
+[scenario]
+type = "particle_resampling"
+name = "extreme-particle-weights"
+seed = 7
+
+[data]
+particles = [[0.0], [1.0]]
+weights = [{max_float!r}, {max_float / 2.0!r}]
+num_samples = 8
+""".strip(),
+        encoding="utf-8",
+    )
+
+    result = run_scenario(scenario)
+
+    assert math.isclose(result.metrics["max_weight"], 2.0 / 3.0)
+    assert math.isclose(result.metrics["min_weight"], 1.0 / 3.0)
+    assert math.isclose(
+        result.metrics["max_weight"] + result.metrics["min_weight"],
+        1.0,
+    )
+    assert math.isfinite(result.metrics["effective_sample_size"])