diff --git a/PreciseNumber.Test/PreciseNumberTests.cs b/PreciseNumber.Test/PreciseNumberTests.cs
index ae3c204..29f8236 100644
--- a/PreciseNumber.Test/PreciseNumberTests.cs
+++ b/PreciseNumber.Test/PreciseNumberTests.cs
@@ -1056,6 +1056,26 @@ public void TestReduceSignificance()
 		Assert.AreEqual(number, number.ReduceSignificance(5));
 	}
 
+	// A precision below one has no meaning, and used to return a wrong number rather than fail:
+	// 5 reduced to zero digits came back as 10, and to minus one digit as 0.
+
+	[TestMethod]
+	[DataRow(0)]
+	[DataRow(-1)]
+	[DataRow(int.MinValue)]
+	public void TestReduceSignificanceRejectsAPrecisionBelowOne(int significantDigits)
+	{
+		Assert.ThrowsExactly<ArgumentOutOfRangeException>(() => 5.ToPreciseNumber().ReduceSignificance(significantDigits));
+		Assert.ThrowsExactly<ArgumentOutOfRangeException>(() => PreciseNumber.Pi.ReduceSignificance(significantDigits));
+	}
+
+	[TestMethod]
+	public void TestReduceSignificanceThrowsWhenTheExponentLeavesTheIntRange()
+	{
+		PreciseNumber number = PreciseNumber.CreateFromComponents(int.MaxValue - 1, 12345);
+		Assert.ThrowsExactly<OverflowException>(() => number.ReduceSignificance(3));
+	}
+
 	[TestMethod]
 	public void TestReduceSignificanceRoundsHalfAwayFromZeroOnTheDroppedDigits()
 	{
diff --git a/PreciseNumber/PreciseNumber.cs b/PreciseNumber/PreciseNumber.cs
index 68184fb..520f46c 100644
--- a/PreciseNumber/PreciseNumber.cs
+++ b/PreciseNumber/PreciseNumber.cs
@@ -857,8 +857,15 @@ internal long CountDecimalDigits() =>
 	/// <param name="significantDigits">The number of significant digits to reduce to.</param>
 	/// <returns>A new instance of <see cref="PreciseNumber"/> reduced to the specified number of significant digits.</returns>
 	/// <remarks>Rounds half away from zero, so 123.5 becomes 124 and 123.456 becomes 123 at three digits.</remarks>
+	/// <exception cref="ArgumentOutOfRangeException">Thrown when <paramref name="significantDigits"/> is less than one.</exception>
+	/// <exception cref="OverflowException">Thrown when the result needs an exponent outside the range of an <see cref="int"/>.</exception>
 	public PreciseNumber ReduceSignificance(int significantDigits)
 	{
+		if (significantDigits < 1)
+		{
+			throw new ArgumentOutOfRangeException(nameof(significantDigits), significantDigits, "At least one significant digit is required.");
+		}
+
 		int significantDifference = significantDigits < SignificantDigits
 			? SignificantDigits - significantDigits
 			: 0;
@@ -868,9 +875,7 @@ public PreciseNumber ReduceSignificance(int significantDigits)
 			return this;
 		}
 
-		int newExponent = Exponent == 0
-			? significantDifference
-			: Exponent + significantDifference;
+		int newExponent = checked(Exponent + significantDifference);
 		BigInteger newSignificand = DropDigitsRoundingHalfAwayFromZero(Significand, significantDifference);
 		return new(newExponent, newSignificand);
 	}