diff --git a/base/float.jl b/base/float.jl
index fc4cef09b48ad..aa61f2c8449e3 100644
--- a/base/float.jl
+++ b/base/float.jl
@@ -882,7 +882,10 @@ for Ti in (Int8, Int16, Int32, Int64, Int128, UInt8, UInt16, UInt32, UInt64, UIn
                     end
                 end
                 function (::Type{$Ti})(x::$Tf)
-                    if ($(Tf(typemin(Ti))) <= x <= $(Tf(typemax(Ti)))) && isinteger(x)
+                    # When typemax(Ti) is not representable by Tf but typemax(Ti) + 1 is,
+                    # then < Tf(typemax(Ti) + 1) is stricter than <= Tf(typemax(Ti)). Using
+                    # the former causes us to throw on UInt64(Float64(typemax(UInt64))+1)
+                    if ($(Tf(typemin(Ti))) <= x < $(Tf(typemax(Ti))+one(Tf))) && isinteger(x)
                         return unsafe_trunc($Ti,x)
                     else
                         throw(InexactError($(Expr(:quote,Ti.name.name)), $Ti, x))
diff --git a/test/floatfuncs.jl b/test/floatfuncs.jl
index a0d6e8743514b..268d699684cbb 100644
--- a/test/floatfuncs.jl
+++ b/test/floatfuncs.jl
@@ -256,3 +256,34 @@ end
         @test isapprox(typemin(T), 0.0, rtol=1)
     end
 end
+
+@testset "Conversion from floating point to unsigned integer near extremes (#51063)" begin
+    @test_throws InexactError UInt32(4.2949673f9)
+    @test_throws InexactError UInt64(1.8446744f19)
+    @test_throws InexactError UInt64(1.8446744073709552e19)
+    @test_throws InexactError UInt128(3.402823669209385e38)
+end
+
+@testset "Conversion from floating point to integer near extremes (exhaustive)" begin
+    for Ti in Base.BitInteger_types, Tf in (Float16, Float32, Float64), x in (typemin(Ti), typemax(Ti))
+        y = Tf(x)
+        for i in -3:3
+            z = nextfloat(y, i)
+
+            result = isfinite(z) ? round(BigInt, z) : error
+            result = result !== error && typemin(Ti) <= result <= typemax(Ti) ? result : error
+
+            if result === error
+                # @test_throws InexactError round(Ti, z) Broken because of #51113
+                @test_throws InexactError Ti(z)
+            else
+                @test result == round(Ti, z)
+                if isinteger(z)
+                    @test result == Ti(z)
+                else
+                    @test_throws InexactError Ti(z)
+                end
+            end
+        end
+    end
+end