diff --git a/src/sparsematrix.jl b/src/sparsematrix.jl
index 56ee216a..93bb439d 100644
--- a/src/sparsematrix.jl
+++ b/src/sparsematrix.jl
@@ -2237,9 +2237,19 @@ Base.iszero(A::AbstractSparseMatrixCSC) = iszero(nzvalview(A))
 function Base.isone(A::AbstractSparseMatrixCSC)
     m, n = size(A)
     m == n && getcolptr(A)[n+1] >= n+1 || return false
-    for j in axes(A,2), k in getcolptr(A)[j]:(getcolptr(A)[j+1] - 1)
-        i, x = rowvals(A)[k], nonzeros(A)[k]
-        ifelse(i == j, isone(x), iszero(x)) || return false
+    for j in axes(A,2)
+        founddiag = false
+        for k in getcolptr(A)[j]:(getcolptr(A)[j+1] - 1)
+            i, x = rowvals(A)[k], nonzeros(A)[k]
+            if i == j
+                isone(x) || return false
+                founddiag = true
+            else
+                iszero(x) || return false
+            end
+        end
+        # every column must have a stored diagonal entry equal to one
+        founddiag || return false
     end
     return true
 end
diff --git a/test/sparsematrix_ops.jl b/test/sparsematrix_ops.jl
index ab2065cb..ee254dce 100644
--- a/test/sparsematrix_ops.jl
+++ b/test/sparsematrix_ops.jl
@@ -46,6 +46,12 @@ end
     @test !isone(spzeros(3, 3))     # test failure for too few stored entries
     @test !isone(sparse(2I, 3, 3))  # test failure for non-one diagonal entries
     @test !isone(sparse(Bidiagonal(fill(1, 3), fill(1, 2), :U))) # test failure for non-zero off-diag entries
+    # issue #763: stored zeros must not be counted towards the diagonal
+    M = sparse([1 0; 1 1]) * sparse([1 0; -1 0])
+    @test nnz(M) == 2 && !isone(M) && !isone(Matrix(M))
+    @test !isone(SparseMatrixCSC(2, 2, [1, 3, 3], [1, 2], [1, 0]))
+    @test !isone(SparseMatrixCSC(2, 2, [1, 2, 3], [1, 1], [1, 0]))
+    @test isone(SparseMatrixCSC(2, 2, [1, 3, 4], [1, 2, 2], [1, 0, 1]))  # stored zero off-diagonal is fine
 end
 
 @testset "indtype" begin